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<h2 id="historical-development">Historical development</h2> <p>In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand, <a href="/facts/Bernard_Bolzano/BUbRVY4p">Bernard Bolzano</a> (1817) had been aware that any bounded sequence of points (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some other point, called a <a href="/facts/Limit_point_of_a_sequence/Ktge2RwL">limit point</a>. Bolzano's proof relied on the <a href="/facts/Method_of_bisection/3tWmtoIp">method of bisection</a>: the sequence was placed into an interval that was then divided into two equal parts, and a part containing infinitely many terms of the sequence was selected. The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts – until it closes down on the desired limit point. The full significance of <a href="/facts/Bolzano%E2%80%93Weierstrass_theorem/3h7gu7r2">Bolzano's theorem</a>, and its method of proof, would not emerge until almost 50 years later when it was rediscovered by <a href="/facts/Karl_Weierstrass/4U8vdQzL">Karl Weierstrass</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p>In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for <a href="/facts/Function_space/lLUE2R1t">spaces of functions</a> rather than just numbers or geometrical points. The idea of regarding functions as themselves points of a generalized space dates back to the investigations of <a href="/facts/Giulio_Ascoli/wNJpSsJx">Giulio Ascoli</a> and <a href="/facts/Cesare_Arzel%C3%A0/a6V17UI2">Cesare Arzelà</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> The culmination of their investigations, the <a href="/facts/Arzel%C3%A0%E2%80%93Ascoli_theorem/eqSFHXbo">Arzelà–Ascoli theorem</a>, was a generalization of the Bolzano–Weierstrass theorem to families of <a href="/facts/Continuous_function/sKbl02pB">continuous functions</a>, the precise conclusion of which was that it was possible to extract a <a href="/facts/Uniform_convergence/ecz9eNWn">uniformly convergent</a> sequence of functions from a suitable family of functions. The uniform limit of this sequence then played precisely the same role as Bolzano's "limit point". Towards the beginning of the twentieth century, results similar to that of Arzelà and Ascoli began to accumulate in the area of <a href="/facts/Integral_equation/LCu4Imfq">integral equations</a>, as investigated by <a href="/facts/David_Hilbert/1vhlHn4b">David Hilbert</a> and <a href="/facts/Erhard_Schmidt/5UGgwTah">Erhard Schmidt</a>. For a certain class of <a href="/facts/Green%27s_functions/rKxKKPcp">Green's functions</a> coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà–Ascoli theorem held in the sense of <a href="/facts/Mean_convergence/pWdatFlY">mean convergence</a> – or convergence in what would later be dubbed a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a>. This ultimately led to the notion of a <a href="/facts/Compact_operator/KNkcK4mE">compact operator</a> as an offshoot of the general notion of a compact space. It was <a href="/facts/Maurice_Ren%C3%A9_Fr%C3%A9chet/QXrzxA1s">Maurice Fréchet</a> who, in 1906, had distilled the essence of the Bolzano–Weierstrass property and coined the term <i>compactness</i> to refer to this general phenomenon (he used the term already in his 1904 paper<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> which led to the famous 1906 thesis). </p><p>However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the <a href="/facts/Linear_continuum/BHs0Wn5m">continuum</a>, which was seen as fundamental for the rigorous formulation of analysis. In 1870, <a href="/facts/Eduard_Heine/FY8enC39">Eduard Heine</a> showed that a <a href="/facts/Continuous_function/sKbl02pB">continuous function</a> defined on a closed and bounded interval was in fact <a href="/facts/Uniformly_continuous/hjvw7tII">uniformly continuous</a>. In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it. The significance of this lemma was recognized by <a href="/facts/%C3%89mile_Borel/lF9PBa1T">Émile Borel</a> (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895) and <a href="/facts/Henri_Lebesgue/ysngVGXc">Henri Lebesgue</a> (1904). The <a href="/facts/Heine%E2%80%93Borel_theorem/c74qCFHY">Heine–Borel theorem</a>, as the result is now known, is another special property possessed by closed and bounded sets of real numbers. </p><p>This property was significant because it allowed for the passage from <a href="/facts/Local_property/sdBEwIy2">local information</a> about a set (such as the continuity of a function) to global information about the set (such as the uniform continuity of a function). This sentiment was expressed by Lebesgue (1904), who also exploited it in the development of the <a href="/facts/Lebesgue_integral/f4AgvC6x">integral now bearing his name</a>. Ultimately, the Russian school of <a href="/facts/Point-set_topology/ezC7B0yP">point-set topology</a>, under the direction of <a href="/facts/Pavel_Alexandrov/booz6U09">Pavel Alexandrov</a> and <a href="/facts/Pavel_Urysohn/fAPrMKzJ">Pavel Urysohn</a>, formulated Heine–Borel compactness in a way that could be applied to the modern notion of a <a href="/facts/Topological_space/v2ut7CbK">topological space</a>. Alexandrov & Urysohn (1929) showed that the earlier version of compactness due to Fréchet, now called (relative) <a href="/facts/Sequential_compactness/5vL96D9l">sequential compactness</a>, under appropriate conditions followed from the version of compactness that was formulated in terms of the existence of finite subcovers. It was this notion of compactness that became the dominant one, because it was not only a stronger property, but it could be formulated in a more general setting with a minimum of additional technical machinery, as it relied only on the structure of the open sets in a space. </p> <h2 id="basic-examples">Basic examples</h2> <p>Any <a href="/facts/Finite_topological_space/HKD3e0qp">finite space</a> is compact; a finite subcover can be obtained by selecting, for each point, an open set containing it. A nontrivial example of a compact space is the (closed) <a href="/facts/Unit_interval/W4dP8IJo">unit interval</a> [0,1] of <a href="/facts/Real_number/R02gw5Pb">real numbers</a>. If one chooses an infinite number of distinct points in the unit interval, then there must be some <a href="/facts/Accumulation_point/95MykoGU">accumulation point</a> among these points in that interval. For instance, the odd-numbered terms of the sequence 1, 1/2, 1/3, 3/4, 1/5, 5/6, 1/7, 7/8, ... get arbitrarily close to 0, while the even-numbered ones get arbitrarily close to 1. The given example sequence shows the importance of including the <a href="/facts/Boundary_(topology)/WgdH18kp">boundary</a> points of the interval, since the <a href="/facts/Limit_of_a_sequence/Ktge2RwL">limit points</a> must be in the space itself — an open (or half-open) interval of the real numbers is not compact. It is also crucial that the interval be <a href="/facts/Bounded_set/yCldjtoy">bounded</a>, since in the interval [0,∞), one could choose the sequence of points 0, 1, 2, 3, ..., of which no sub-sequence ultimately gets arbitrarily close to any given real number. </p><p>In two dimensions, closed <a href="/facts/Disk_(mathematics)/DgMVr97H">disks</a> are compact since for any infinite number of points sampled from a disk, some subset of those points must get arbitrarily close either to a point within the disc, or to a point on the boundary. However, an open disk is not compact, because a sequence of points can tend to the boundary – without getting arbitrarily close to any point in the interior. Likewise, spheres are compact, but a sphere missing a point is not since a sequence of points can still tend to the missing point, thereby not getting arbitrarily close to any point <i>within</i> the space. Lines and planes are not compact, since one can take a set of equally-spaced points in any given direction without approaching any point. </p> <h2 id="definitions">Definitions</h2> <p>Various definitions of compactness may apply, depending on the level of generality. A subset of <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> in particular is called compact if it is <a href="/facts/Closed_set/upYnRTUj">closed</a> and <a href="/facts/Bounded_set/yCldjtoy">bounded</a>. This implies, by the <a href="/facts/Bolzano%E2%80%93Weierstrass_theorem/3h7gu7r2">Bolzano–Weierstrass theorem</a>, that any infinite <a href="/facts/Sequence_(mathematics)/gV2S4uqp">sequence</a> from the set has a <a href="/facts/Subsequence/GdC2Qwau">subsequence</a> that converges to a point in the set. Various equivalent notions of compactness, such as <a href="/facts/Sequential_compactness/5vL96D9l">sequential compactness</a> and <a href="/facts/Limit_point_compact/tB8tbHJV">limit point compactness</a>, can be developed in general <a href="/facts/Metric_space/gnBBRXtc">metric spaces</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p><p>In contrast, the different notions of compactness are not equivalent in general <a href="/facts/Topological_space/v2ut7CbK">topological spaces</a>, and the most useful notion of compactness – originally called <i>bicompactness</i> – is defined using <a href="/facts/Cover_(topology)/2VUdelA4">covers</a> consisting of <a href="/facts/Open_set/QfTCIqMu">open sets</a> (see <i>Open cover definition</i> below). That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the <a href="/facts/Heine%E2%80%93Borel_theorem/c74qCFHY">Heine–Borel theorem</a>. Compactness, when defined in this manner, often allows one to take information that is known <a href="/facts/Local_property/sdBEwIy2">locally</a> – in a <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighbourhood</a> of each point of the space – and to extend it to information that holds globally throughout the space. An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is <a href="/facts/Uniformly_continuous/hjvw7tII">uniformly continuous</a>; here, continuity is a local property of the function, and uniform continuity the corresponding global property. </p> <h3>Open cover definition</h3> <p>Formally, a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> X is called <i>compact</i> if every <a href="/facts/Open_cover/2VUdelA4">open cover</a> of X has a <a href="/facts/Finite_set/za1newlP">finite</a> <a href="/facts/Subcover/2VUdelA4">subcover</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> That is, X is compact if for every collection C of open subsets<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> of X such that </p><p> X = ⋃ S ∈ C S , {\displaystyle X=\bigcup _{S\in C}S\ ,} </p><p>there is a finite subcollection F ⊆ C such that </p><p> X = ⋃ S ∈ F S . {\displaystyle X=\bigcup _{S\in F}S\ .} </p><p>Some branches of mathematics such as <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, typically influenced by the French school of <a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki</a>, use the term <i>quasi-compact</i> for the general notion, and reserve the term <i>compact</i> for topological spaces that are both <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> and <i>quasi-compact</i>. A compact set is sometimes referred to as a <i>compactum</i>, plural <i>compacta</i>. </p> <h3>Compactness of subsets</h3> <p>A subset K of a topological space X is said to be compact if it is compact as a subspace (in the <a href="/facts/Subspace_topology/NjU6z4Ca">subspace topology</a>). That is, K is compact if for every arbitrary collection C of open subsets of X such that </p><p> K ⊆ ⋃ S ∈ C S , {\displaystyle K\subseteq \bigcup _{S\in C}S\ ,} </p><p>there is a finite subcollection F ⊆ C such that </p><p> K ⊆ ⋃ S ∈ F S . {\displaystyle K\subseteq \bigcup _{S\in F}S\ .} </p><p>Because compactness is a <a href="/facts/Topological_property/vbJR18Sx">topological property</a>, the compactness of a subset depends only on the subspace topology induced on it. It follows that, if K ⊂ Z ⊂ Y {\displaystyle K\subset Z\subset Y} , with subset Z equipped with the subspace topology, then K is compact in Z if and only if K is compact in Y. </p> <h3>Characterization</h3> <p>If X is a topological space then the following are equivalent: </p> <ol><li>X is compact; i.e., every <a href="/facts/Open_cover/2VUdelA4">open cover</a> of X has a finite <a href="/facts/Subcover/2VUdelA4">subcover</a>.</li> <li>X has a sub-base such that every cover of the space, by members of the sub-base, has a finite subcover (<a href="/facts/Alexander%27s_sub-base_theorem/CoQAlDFJ">Alexander's sub-base theorem</a>).</li> <li>X is <a href="/facts/Lindel%C3%B6f_space/HC4t6bvz">Lindelöf</a> and <a href="/facts/Countably_compact/Ppm9uXq5">countably compact</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a></li> <li>Any collection of closed subsets of X with the <a href="/facts/Finite_intersection_property/xg5uX9n0">finite intersection property</a> has nonempty intersection.</li> <li>Every <a href="/facts/Net_(mathematics)/msmsvMTT">net</a> on X has a convergent subnet (see the article on <a href="/facts/Net_(mathematics)/msmsvMTT">nets</a> for a proof).</li> <li>Every <a href="/facts/Filters_in_topology/5vQbHcpE">filter</a> on X has a convergent refinement.</li> <li>Every net on X has a cluster point.</li> <li>Every filter on X has a cluster point.</li> <li>Every <a href="/facts/Ultrafilter_(set_theory)/KUpERdqN">ultrafilter</a> on X converges to at least one point.</li> <li>Every infinite subset of X has a <a href="/facts/Complete_accumulation_point/95MykoGU">complete accumulation point</a>.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a></li> <li>For every topological space Y, the projection X × Y → Y {\displaystyle X\times Y\to Y} is a <a href="/facts/Closed_mapping/abHUKX7l">closed mapping</a><a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> (see <a href="/facts/Proper_map/jJS4YK7o">proper map</a>).</li> <li>Every open cover linearly ordered by subset inclusion contains X.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a></li></ol> <p>Bourbaki defines a compact space (quasi-compact space) as a topological space where each filter has a cluster point (i.e., 8. in the above).<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p> <h4>Euclidean space</h4> <p>For any <a href="/facts/Subset/SJGERmbA">subset</a> A of <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>, A is compact if and only if it is <a href="/facts/Closed_set/upYnRTUj">closed</a> and <a href="/facts/Bounded_set/yCldjtoy">bounded</a>; this is the <a href="/facts/Heine%E2%80%93Borel_theorem/c74qCFHY">Heine–Borel theorem</a>. </p><p>As a <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> is a metric space, the conditions in the next subsection also apply to all of its subsets. Of all of the equivalent conditions, it is in practice easiest to verify that a subset is closed and bounded, for example, for a closed <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> or closed n-ball. </p> <h4>Metric spaces</h4> <p>For any metric space (<i>X</i>, <i>d</i>), the following are equivalent (assuming <a href="/facts/Countable_choice/goSPxEdj">countable choice</a>): </p> <ol><li>(<i>X</i>, <i>d</i>) is compact.</li> <li>(<i>X</i>, <i>d</i>) is <a href="/facts/Completeness_(topology)/lsckmU8G">complete</a> and <a href="/facts/Totally_bounded/H8GSU1TT">totally bounded</a> (this is also equivalent to compactness for <a href="/facts/Uniform_space/GKumHlHo">uniform spaces</a>).<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a></li> <li>(<i>X</i>, <i>d</i>) is sequentially compact; that is, every <a href="/facts/Sequence/gV2S4uqp">sequence</a> in X has a convergent subsequence whose limit is in X (this is also equivalent to compactness for <a href="/facts/First-countable/HDuTYa6A">first-countable</a> <a href="/facts/Uniform_space/GKumHlHo">uniform spaces</a>).</li> <li>(<i>X</i>, <i>d</i>) is <a href="/facts/Limit_point_compact/tB8tbHJV">limit point compact</a> (also called weakly countably compact); that is, every infinite subset of X has at least one <a href="/facts/Limit_point_of_a_set/95MykoGU">limit point</a> in X.</li> <li>(<i>X</i>, <i>d</i>) is <a href="/facts/Countably_compact/Ppm9uXq5">countably compact</a>; that is, every countable open cover of X has a finite subcover.</li> <li>(<i>X</i>, <i>d</i>) is an image of a continuous function from the <a href="/facts/Cantor_set/6NUcMJvN">Cantor set</a>.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a></li> <li>Every decreasing nested sequence of nonempty closed subsets <i>S</i>1 ⊇ <i>S</i>2 ⊇ ... in (<i>X</i>, <i>d</i>) has a nonempty intersection.</li> <li>Every increasing nested sequence of proper open subsets <i>S</i>1 ⊆ <i>S</i>2 ⊆ ... in (<i>X</i>, <i>d</i>) fails to cover X.</li></ol> <p>A compact metric space (<i>X</i>, <i>d</i>) also satisfies the following properties: </p> <ol><li><a href="/facts/Lebesgue%27s_number_lemma/EPaRuWyT">Lebesgue's number lemma</a>: For every open cover of X, there exists a number <i>δ</i> > 0 such that every subset of X of diameter < δ is contained in some member of the cover.</li> <li>(<i>X</i>, <i>d</i>) is <a href="/facts/Second-countable_space/qw9YdnBN">second-countable</a>, <a href="/facts/Separable_space/2bmU73bk">separable</a> and <a href="/facts/Lindel%C3%B6f_space/HC4t6bvz">Lindelöf</a> – these three conditions are equivalent for metric spaces. The converse is not true; e.g., a countable discrete space satisfies these three conditions, but is not compact.</li> <li>X is closed and bounded (as a subset of any metric space whose restricted metric is d). The converse may fail for a non-Euclidean space; e.g. the <a href="/facts/Real_line/6eq42xX5">real line</a> equipped with the <a href="/facts/Discrete_metric/hpvXA23u">discrete metric</a> is closed and bounded but not compact, as the collection of all <a href="/facts/Singleton_(mathematics)/LtNVfujT">singletons</a> of the space is an open cover which admits no finite subcover. It is complete but not totally bounded.</li></ol> <h4>Ordered spaces</h4> <p>For an ordered space (<i>X</i>, <) (i.e. a totally ordered set equipped with the order topology), the following are equivalent: </p> <ol><li>(<i>X</i>, <) is compact.</li> <li>Every subset of X has a supremum (i.e. a least upper bound) in X.</li> <li>Every subset of X has an infimum (i.e. a greatest lower bound) in X.</li> <li>Every nonempty closed subset of X has a maximum and a minimum element.</li></ol> <p>An ordered space satisfying (any one of) these conditions is called a complete lattice. </p><p>In addition, the following are equivalent for all ordered spaces (<i>X</i>, <), and (assuming <a href="/facts/Countable_choice/goSPxEdj">countable choice</a>) are true whenever (<i>X</i>, <) is compact. (The converse in general fails if (<i>X</i>, <) is not also metrizable.): </p> <ol><li>Every sequence in (<i>X</i>, <) has a subsequence that converges in (<i>X</i>, <).</li> <li>Every monotone increasing sequence in X converges to a unique limit in X.</li> <li>Every monotone decreasing sequence in X converges to a unique limit in X.</li> <li>Every decreasing nested sequence of nonempty closed subsets S1 ⊇ S2 ⊇ ... in (<i>X</i>, <) has a nonempty intersection.</li> <li>Every increasing nested sequence of proper open subsets S1 ⊆ S2 ⊆ ... in (<i>X</i>, <) fails to cover X.</li></ol> <h4>Characterization by continuous functions</h4> <p>Let X be a topological space and C(<i>X</i>) the ring of real continuous functions on X. For each <i>p</i> ∈ <i>X</i>, the evaluation map ev p : C ( X ) → R {\displaystyle \operatorname {ev} _{p}\colon C(X)\to \mathbb {R} } given by ev<i>p</i>(<i>f</i>) = <i>f</i>(<i>p</i>) is a ring homomorphism. The <a href="/facts/Kernel_(algebra)/GGSpUPMJ">kernel</a> of ev<i>p</i> is a <a href="/facts/Maximal_ideal/nf4DnKVS">maximal ideal</a>, since the <a href="/facts/Residue_field/e6nflEmB">residue field</a> C(<i>X</i>)/ker ev<i>p</i> is the field of real numbers, by the <a href="/facts/First_isomorphism_theorem/SNpPTkrd">first isomorphism theorem</a>. A topological space X is <a href="/facts/Pseudocompact_space/Itcq221R">pseudocompact</a> if and only if every maximal ideal in C(<i>X</i>) has residue field the real numbers. For <a href="/facts/Completely_regular_space/jyCgpKB5">completely regular spaces</a>, this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> There are pseudocompact spaces that are not compact, though. </p><p>In general, for non-pseudocompact spaces there are always maximal ideals m in C(<i>X</i>) such that the residue field C(<i>X</i>)/<i>m</i> is a (<a href="/facts/Non-archimedean_field/61pC7UP6">non-Archimedean</a>) <a href="/facts/Hyperreal_field/Kvf09qpo">hyperreal field</a>. The framework of <a href="/facts/Non-standard_analysis/lEpNpWMB">non-standard analysis</a> allows for the following alternative characterization of compactness:<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> a topological space X is compact if and only if every point x of the natural extension <i>*X</i> is <a href="/facts/Infinitesimal/JKcTMIgD">infinitely close</a> to a point <i>x</i>0 of X (more precisely, x is contained in the <a href="/facts/Monad_(non-standard_analysis)/pJ3DspXx">monad</a> of <i>x</i>0). </p> <h4>Hyperreal definition</h4> <p>A space X is compact if its <a href="/facts/Hyperreal_number/Kvf09qpo">hyperreal extension</a> <i>*X</i> (constructed, for example, by the <a href="/facts/Ultrapower_construction/Kvf09qpo">ultrapower construction</a>) has the property that every point of <i>*X</i> is infinitely close to some point of <i>X</i> ⊂ <i>*X</i>. For example, an open real interval <i>X</i> = (0, 1) is not compact because its hyperreal extension *(0,1) contains infinitesimals, which are infinitely close to 0, which is not a point of X. </p> <h2 id="sufficient-conditions">Sufficient conditions</h2> <ul><li>A closed subset of a compact space is compact.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a></li> <li>A finite <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> of compact sets is compact.</li> <li>A <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous</a> image of a compact space is compact.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a></li> <li>The intersection of any non-empty collection of compact subsets of a Hausdorff space is compact (and closed); <ul><li>If X is not Hausdorff then the intersection of two compact subsets may fail to be compact (see footnote for example).<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a></li></ul></li> <li>The <a href="/facts/Product_topology/0hg02IaX">product</a> of any collection of compact spaces is compact. (This is <a href="/facts/Tychonoff%27s_theorem/PhVArhKr">Tychonoff's theorem</a>, which is equivalent to the <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a>.)</li> <li>In a <a href="/facts/Metrizable_space/BCLdo5Jf">metrizable space</a>, a subset is compact if and only if it is <a href="/facts/Sequentially_compact/5vL96D9l">sequentially compact</a> (assuming <a href="/facts/Axiom_of_countable_choice/goSPxEdj">countable choice</a>)</li> <li>A finite set endowed with any topology is compact.</li></ul> <h2 id="properties-of-compact-spaces">Properties of compact spaces</h2> <ul><li>A compact subset of a <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff space</a> X is closed. <ul><li>If X is not Hausdorff then a compact subset of X may fail to be a closed subset of X (see footnote for example).<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a></li> <li>If X is not Hausdorff then the closure of a compact set may fail to be compact (see footnote for example).<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a></li></ul></li> <li>In any <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> (TVS), a compact subset is <a href="/facts/Complete_space/lsckmU8G">complete</a>. However, every non-Hausdorff TVS contains compact (and thus complete) subsets that are <i>not</i> closed.</li> <li>If A and B are disjoint compact subsets of a Hausdorff space X, then there exist disjoint open sets U and V in X such that <i>A</i> ⊆ <i>U</i> and <i>B</i> ⊆ <i>V</i>.</li> <li>A continuous bijection from a compact space into a Hausdorff space is a <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a>.</li> <li>A compact Hausdorff space is <a href="/facts/Normal_space/t4DLqLCd">normal</a> and <a href="/facts/Regular_space/gLvRIhRH">regular</a>.</li> <li>If a space X is compact and Hausdorff, then no finer topology on X is compact and no coarser topology on X is Hausdorff.</li> <li>If a subset of a metric space (<i>X</i>, <i>d</i>) is compact then it is d-bounded.</li></ul> <h3>Functions and compact spaces</h3> <p>Since a <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous</a> image of a compact space is compact, the <a href="/facts/Extreme_value_theorem/eSV6P7eh">extreme value theorem</a> holds for such spaces: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum.<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> (Slightly more generally, this is true for an upper semicontinuous function.) As a sort of converse to the above statements, the pre-image of a compact space under a <a href="/facts/Proper_map/jJS4YK7o">proper map</a> is compact. </p> <h3>Compactifications</h3> <p class="note">Main article: <a href="/facts/Compactification_(mathematics)/Te8RASyR">Compactification (mathematics)</a></p> <p>Every topological space X is an open <a href="/facts/Dense_topological_subspace/NjU6z4Ca">dense subspace</a> of a compact space having at most one point more than X, by the <a href="/facts/Alexandroff_one-point_compactification/d3Xc0QKu">Alexandroff one-point compactification</a>. By the same construction, every <a href="/facts/Locally_compact/hesalT7q">locally compact</a> Hausdorff space X is an open dense subspace of a compact Hausdorff space having at most one point more than X. </p> <h3>Ordered compact spaces</h3> <p>A nonempty compact subset of the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> has a greatest element and a least element. </p><p>Let X be a <a href="/facts/Total_order/xnTtmuYJ">simply ordered</a> set endowed with the <a href="/facts/Order_topology/EHISytz6">order topology</a>. Then X is compact if and only if X is a <a href="/facts/Complete_lattice/pQbphWeR">complete lattice</a> (i.e. all subsets have suprema and infima).<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> </p> <h2 id="examples">Examples</h2> <ul><li>Any <a href="/facts/Finite_topological_space/HKD3e0qp">finite topological space</a>, including the <a href="/facts/Empty_set/ffEk3eA6">empty set</a>, is compact. More generally, any space with a <a href="/facts/Finite_topology/JjXlyXdE">finite topology</a> (only finitely many open sets) is compact; this includes in particular the <a href="/facts/Trivial_topology/uA7RlGth">trivial topology</a>.</li> <li>Any space carrying the <a href="/facts/Cofinite_topology/LvpB23pu">cofinite topology</a> is compact.</li> <li>Any <a href="/facts/Locally_compact/hesalT7q">locally compact</a> Hausdorff space can be turned into a compact space by adding a single point to it, by means of <a href="/facts/Alexandroff_one-point_compactification/d3Xc0QKu">Alexandroff one-point compactification</a>. The one-point compactification of R {\displaystyle \mathbb {R} } is homeomorphic to the circle S1; the one-point compactification of R 2 {\displaystyle \mathbb {R} ^{2}} is homeomorphic to the sphere S2. Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.</li> <li>The <a href="/facts/Right_order_topology/EHISytz6">right order topology</a> or <a href="/facts/Left_order_topology/EHISytz6">left order topology</a> on any bounded <a href="/facts/Totally_ordered_set/xnTtmuYJ">totally ordered set</a> is compact. In particular, <a href="/facts/Sierpi%C5%84ski_space/Ha6nckJV">Sierpiński space</a> is compact.</li> <li>No <a href="/facts/Discrete_space/hpvXA23u">discrete space</a> with an infinite number of points is compact. The collection of all <a href="/facts/Singleton_(mathematics)/LtNVfujT">singletons</a> of the space is an open cover which admits no finite subcover. Finite discrete spaces are compact.</li> <li>In R {\displaystyle \mathbb {R} } carrying the <a href="/facts/Lower_limit_topology/uE1nTKxG">lower limit topology</a>, no uncountable set is compact.</li> <li>In the <a href="/facts/Cocountable_topology/NyL1pumR">cocountable topology</a> on an uncountable set, no infinite set is compact. Like the previous example, the space as a whole is not <a href="/facts/Locally_compact/hesalT7q">locally compact</a> but is still <a href="/facts/Lindel%C3%B6f_space/HC4t6bvz">Lindelöf</a>.</li> <li>The closed <a href="/facts/Unit_interval/W4dP8IJo">unit interval</a> [0, 1] is compact. This follows from the <a href="/facts/Heine%E2%80%93Borel_theorem/c74qCFHY">Heine–Borel theorem</a>. The open interval (0, 1) is not compact: the <a href="/facts/Open_cover/2VUdelA4">open cover</a> ( 1 n , 1 − 1 n ) {\textstyle \left({\frac {1}{n}},1-{\frac {1}{n}}\right)} for n = 3, 4, ... does not have a finite subcover. Similarly, the set of <i><a href="/facts/Rational_numbers/VJQmyY05">rational numbers</a></i> in the closed interval [0,1] is not compact: the sets of rational numbers in the intervals [ 0 , 1 π − 1 n ] and [ 1 π + 1 n , 1 ] {\textstyle \left[0,{\frac {1}{\pi }}-{\frac {1}{n}}\right]{\text{ and }}\left[{\frac {1}{\pi }}+{\frac {1}{n}},1\right]} cover all the rationals in [0, 1] for n = 4, 5, ... but this cover does not have a finite subcover. Here, the sets are open in the subspace topology even though they are not open as subsets of R {\displaystyle \mathbb {R} } .</li> <li>The set R {\displaystyle \mathbb {R} } of all real numbers is not compact as there is a cover of open intervals that does not have a finite subcover. For example, intervals (n − 1, n + 1) , where n takes all integer values in Z, cover R {\displaystyle \mathbb {R} } but there is no finite subcover.</li> <li>On the other hand, the <a href="/facts/Extended_real_number_line/65Gs7QS0">extended real number line</a> carrying the analogous topology <i>is</i> compact; note that the cover described above would never reach the points at infinity and thus would <i>not</i> cover the extended real line. In fact, the set has the <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a> to [−1, 1] of mapping each infinity to its corresponding unit and every real number to its sign multiplied by the unique number in the positive part of interval that results in its absolute value when divided by one minus itself, and since homeomorphisms preserve covers, the Heine-Borel property can be inferred.</li> <li>For every <a href="/facts/Natural_number/u65og8JE">natural number</a> n, the <a href="/facts/N-sphere/bFu2S7E2">n-sphere</a> is compact. Again from the Heine–Borel theorem, the closed unit ball of any finite-dimensional <a href="/facts/Normed_vector_space/rmDljfyE">normed vector space</a> is compact. This is not true for infinite dimensions; in fact, a normed vector space is finite-dimensional if and only if its <a href="/facts/Closed_unit_ball/5UeoTcug">closed unit ball</a> is compact.</li> <li>On the other hand, the closed unit ball of the dual of a normed space is compact for the weak-* topology. (<a href="/facts/Alaoglu%27s_theorem/NILk5nDm">Alaoglu's theorem</a>)</li> <li>The <a href="/facts/Cantor_set/6NUcMJvN">Cantor set</a> is compact. In fact, every compact metric space is a continuous image of the Cantor set.</li> <li>Consider the set K of all functions <i>f</i> : R {\displaystyle \mathbb {R} } → [0, 1] from the real number line to the closed unit interval, and define a topology on K so that a sequence { f n } {\displaystyle \{f_{n}\}} in K converges towards <i>f</i> ∈ <i>K</i> if and only if { f n ( x ) } {\displaystyle \{f_{n}(x)\}} converges towards <i>f</i>(<i>x</i>) for all real numbers x. There is only one such topology; it is called the topology of <a href="/facts/Pointwise_convergence/TXGpeyIK">pointwise convergence</a> or the <a href="/facts/Product_topology/0hg02IaX">product topology</a>. Then K is a compact topological space; this follows from the <a href="/facts/Tychonoff_theorem/PhVArhKr">Tychonoff theorem</a>.</li> <li>A subset of the Banach space of real-valued continuous functions on a compact Hausdorff space is relatively compact if and only if it is equicontinuous and pointwise bounded (<a href="/facts/Arzel%C3%A0%E2%80%93Ascoli_theorem/eqSFHXbo">Arzelà–Ascoli theorem</a>).</li> <li>Consider the set K of all functions <i>f</i> : [0, 1] → [0, 1] satisfying the <a href="/facts/Lipschitz_condition/Hw20EPEU">Lipschitz condition</a> |<i>f</i>(<i>x</i>) − <i>f</i>(<i>y</i>)| ≤ |<i>x</i> − <i>y</i>| for all <i>x</i>, <i>y</i> ∈ [0,1]. Consider on K the metric induced by the <a href="/facts/Uniform_convergence/ecz9eNWn">uniform distance</a> d ( f , g ) = sup x ∈ [ 0 , 1 ] | f ( x ) − g ( x ) | . {\displaystyle d(f,g)=\sup _{x\in [0,1]}|f(x)-g(x)|.} Then by the Arzelà–Ascoli theorem the space K is compact.</li> <li>The <a href="/facts/Spectrum_of_an_operator/cA1evTo6">spectrum</a> of any <a href="/facts/Bounded_linear_operator/LYmDTIqR">bounded linear operator</a> on a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> is a nonempty compact subset of the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> C {\displaystyle \mathbb {C} } . Conversely, any compact subset of C {\displaystyle \mathbb {C} } arises in this manner, as the spectrum of some bounded linear operator. For instance, a diagonal operator on the Hilbert space <a href="/facts/Sequence_spaces/2HkcM12K"> ℓ 2 {\displaystyle \ell ^{2}} </a> may have any compact nonempty subset of C {\displaystyle \mathbb {C} } as spectrum.</li> <li>The space of Borel <a href="/facts/Probability_measure/8BjhgoPR">probability measures</a> on a compact Hausdorff space is compact for the <a href="/facts/Vague_topology/jcHas6W8">vague topology</a>, by the Alaoglu theorem.</li> <li>A collection of probability measures on the Borel sets of Euclidean space is called <i>tight</i> if, for any positive epsilon, there exists a compact subset containing all but at most epsilon of the mass of each of the measures. Helly's theorem then asserts that a collection of probability measures is relatively compact for the vague topology if and only if it is tight.</li></ul> <h3>Algebraic examples</h3> <ul><li><a href="/facts/Topological_group/cAgNMMRU">Topological groups</a> such as an <a href="/facts/Orthogonal_group/jKWM1KDM">orthogonal group</a> are compact, while groups such as a <a href="/facts/General_linear_group/B54gNILS">general linear group</a> are not.</li> <li>Since the <a href="/facts/P-adic_numbers/SDgDbkLF">p-adic integers</a> are <a href="/facts/Homeomorphic/jMfWg3Mn">homeomorphic</a> to the Cantor set, they form a compact set.</li> <li>Any <a href="/facts/Global_field/na458xWs">global field</a> <i>K</i> is a discrete additive subgroup of its <a href="/facts/Adele_ring/2LsbgeBE">adele ring</a>, and the quotient space is compact. This was used in <a href="/facts/John_Tate_(mathematician)/nBntLCG0">John Tate</a>'s <a href="/facts/Tate%27s_thesis/uKKC15um">thesis</a> to allow <a href="/facts/Harmonic_analysis/CyG5bIPb">harmonic analysis</a> to be used in <a href="/facts/Number_theory/zhoa7zrc">number theory</a>.</li> <li>The <a href="/facts/Spectrum_of_a_ring/2k2nDE5w">spectrum</a> of any <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> with the <a href="/facts/Zariski_topology/uolkKBpM">Zariski topology</a> (that is, the set of all prime ideals) is compact, but never <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> (except in trivial cases). In algebraic geometry, such topological spaces are examples of quasi-compact <a href="/facts/Scheme_(mathematics)/c7uSqGqL">schemes</a>, "quasi" referring to the non-Hausdorff nature of the topology.</li> <li>The spectrum of a <a href="/facts/Boolean_algebra/VkCToAmg">Boolean algebra</a> is compact, a fact which is part of the <a href="/facts/Stone_representation_theorem/tN870P19">Stone representation theorem</a>. <a href="/facts/Stone_space/wqW5xc48">Stone spaces</a>, compact <a href="/facts/Totally_disconnected_space/lTu5Rvyw">totally disconnected</a> Hausdorff spaces, form the abstract framework in which these spectra are studied. Such spaces are also useful in the study of <a href="/facts/Profinite_group/0RX9Da9C">profinite groups</a>.</li> <li>The <a href="/facts/Structure_space/3BTHdpg2">structure space</a> of a commutative unital <a href="/facts/Banach_algebra/3BTHdpg2">Banach algebra</a> is a compact Hausdorff space.</li> <li>The <a href="/facts/Hilbert_cube/5t1btBr7">Hilbert cube</a> is compact, again a consequence of Tychonoff's theorem.</li> <li>A <a href="/facts/Profinite_group/0RX9Da9C">profinite group</a> (e.g. <a href="/facts/Galois_group/A0p35p5c">Galois group</a>) is compact.</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Compactly_generated_space/2A5umYGa">Compactly generated space</a></li> <li><a href="/facts/Compactness_theorem/8tHVkdeK">Compactness theorem</a></li> <li><a href="/facts/Eberlein_compactum/C13FTDS8">Eberlein compactum</a></li> <li><a href="/facts/Exhaustion_by_compact_sets/l2S3BcmM">Exhaustion by compact sets</a></li> <li><a href="/facts/Lindel%C3%B6f_space/HC4t6bvz">Lindelöf space</a></li> <li><a href="/facts/Metacompact_space/QYxP3vsj">Metacompact space</a></li> <li><a href="/facts/Noetherian_topological_space/SXazgwJP">Noetherian topological space</a></li> <li><a href="/facts/Orthocompact_space/PbNndhq2">Orthocompact space</a></li> <li><a href="/facts/Paracompact_space/rjTrvF7j">Paracompact space</a></li> <li><a href="/facts/Quasi-compact_morphism/sUCP0wez">Quasi-compact morphism</a></li> <li><a href="/facts/Totally_bounded_space/H8GSU1TT">Precompact set</a> - also called <i><a href="/facts/Totally_bounded/H8GSU1TT">totally bounded</a></i></li> <li><a href="/facts/Relatively_compact_subspace/z5uIqMDp">Relatively compact subspace</a></li> <li><a href="/facts/Totally_bounded/H8GSU1TT">Totally bounded</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="bibliography">Bibliography</h2> <ul><li><a href="/facts/Pavel_Alexandrov/booz6U09">Alexandrov, Pavel</a>; <a href="/facts/Pavel_Urysohn/fAPrMKzJ">Urysohn, Pavel</a> (1929). "Mémoire sur les espaces topologiques compacts". <i>Koninklijke Nederlandse Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Mathematical Sciences</i>. 14.</li> <li>Arkhangel'skii, A.V.; Fedorchuk, V.V. (1990). "The basic concepts and constructions of general topology". In Arkhangel'skii, A.V.; Pontrjagin, L.S. (eds.). <i>General Topology I</i>. Encyclopedia of the Mathematical Sciences. Vol. 17. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-18178-3..</li> <li>Arkhangel'skii, A.V. 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(23 June 1995). <i>Modern Analysis and Topology</i>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>. New York: <a href="/facts/Springer_Publishing/JfRev59b">Springer-Verlag</a> Science & Business Media. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-97986-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/31969970">31969970</a>. <a href="/facts/OL_(identifier)/A13NLNNh">OL</a> <a href="https://openlibrary.org/books/OL1272666M">1272666M</a>.</li> <li>Kelley, John (1955). <i>General topology</i>. Graduate Texts in Mathematics. Vol. 27. Springer-Verlag.</li> <li><a href="/facts/Morris_Kline/kVyuJIh2">Kline, Morris</a> (1990) [1972]. <a href="https://archive.org/details/mathematicalthou00klin"><i>Mathematical thought from ancient to modern times</i></a> (3rd ed.). Oxford University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-506136-9.</li> <li>Lebesgue, Henri (1904). <a href="https://books.google.com/books?id=VfUKAAAAYAAJ&q=%22Lebesgue%22%20%22Le%C3%A7ons%20sur%20l'int%C3%A9gration%20et%20la%20recherche%20des%20fonctions%20...%22&pg=PA1"><i>Leçons sur l'intégration et la recherche des fonctions primitives</i></a>. Gauthier-Villars.</li> <li>Mack, John (1967). "Directed covers and paracompact spaces". <i>Canadian Journal of Mathematics</i>. 19: 649–654. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4153%2FCJM-1967-059-0">10.4153/CJM-1967-059-0</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0211382">0211382</a>.</li> <li><a href="/facts/Abraham_Robinson/UEskqXag">Robinson, Abraham</a> (1996). <i>Non-standard analysis</i>. Princeton University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-04490-3. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0205854">0205854</a>.</li> <li>Scarborough, C.T.; Stone, A.H. (1966). <a href="https://www.ams.org/tran/1966-124-01/S0002-9947-1966-0203679-7/S0002-9947-1966-0203679-7.pdf">"Products of nearly compact spaces"</a> (PDF). <i>Transactions of the American Mathematical Society</i>. 124 (1): 131–147. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1994440">10.2307/1994440</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1994440">1994440</a>. <a href="https://web.archive.org/web/20170816203049/http://www.ams.org/tran/1966-124-01/S0002-9947-1966-0203679-7/S0002-9947-1966-0203679-7.pdf">Archived</a> (PDF) from the original on 2017-08-16..</li> <li><a href="/facts/Lynn_Arthur_Steen/JFdvAfpc">Steen, Lynn Arthur</a>; <a href="/facts/J._Arthur_Seebach,_Jr./sU4TbeUA">Seebach, J. Arthur Jr.</a> (1995) [1978]. <a href="/facts/Counterexamples_in_Topology/f3eAqwaP"><i>Counterexamples in Topology</i></a> (Dover Publications reprint of 1978 ed.). Berlin, New York: Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-68735-3. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0507446">0507446</a>.</li> <li>Willard, Stephen (1970). <i>General Topology</i>. Dover publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-43479-6.</li></ul> <h2 id="external-links">External links</h2> <ul><li>Sundström, Manya Raman (2010). "A pedagogical history of compactness". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1006.4131v1">1006.4131v1</a> [<a href="https://arxiv.org/archive/math.HO">math.HO</a>].</li></ul> <p><i>This article incorporates material from <a href="https://planetmath.org/examplesofcompactspaces">Examples of compact spaces</a> on <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.</i> </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Compactness". Encyclopaedia Britannica. mathematics. Retrieved 2019-11-25 – via britannica.com. <a href="https://www.britannica.com/science/compactness" target="_blank">https://www.britannica.com/science/compactness</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Engelking, Ryszard (1977). General Topology. Warsaw, PL: PWN. p. 266. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>"Sequential compactness". www-groups.mcs.st-andrews.ac.uk. MT 4522 course lectures. Retrieved 2019-11-25. <a href="http://www-groups.mcs.st-andrews.ac.uk/~john/MT4522/Lectures/L22.html" target="_blank">http://www-groups.mcs.st-andrews.ac.uk/~john/MT4522/Lectures/L22.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Kline 1990, pp. 952–953; Boyer & Merzbach 1991, p. 561 - Kline, Morris (1990) [1972]. Mathematical thought from ancient to modern times (3rd ed.). Oxford University Press. ISBN 978-0-19-506136-9. <a href="https://archive.org/details/mathematicalthou00klin" target="_blank">https://archive.org/details/mathematicalthou00klin</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Kline 1990, Chapter 46, §2 - Kline, Morris (1990) [1972]. Mathematical thought from ancient to modern times (3rd ed.). Oxford University Press. ISBN 978-0-19-506136-9. <a href="https://archive.org/details/mathematicalthou00klin" target="_blank">https://archive.org/details/mathematicalthou00klin</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Frechet, M. 1904. "Generalisation d'un theorem de Weierstrass". Analyse Mathematique. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>"Sequential compactness". www-groups.mcs.st-andrews.ac.uk. MT 4522 course lectures. Retrieved 2019-11-25. <a href="http://www-groups.mcs.st-andrews.ac.uk/~john/MT4522/Lectures/L22.html" target="_blank">http://www-groups.mcs.st-andrews.ac.uk/~john/MT4522/Lectures/L22.html</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Weisstein, Eric W. "Compact Space". Wolfram MathWorld. Retrieved 2019-11-25. <a href="http://mathworld.wolfram.com/CompactSpace.html" target="_blank">http://mathworld.wolfram.com/CompactSpace.html</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Here, "collection" means "set" but is used because "collection of open subsets" is less awkward than "set of open subsets". Similarly, "subcollection" means "subset". <a href="/wiki/Set_(mathematics)" target="_blank">/wiki/Set_(mathematics)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Howes 1995, pp. xxvi–xxviii. - Howes, Norman R. (23 June 1995). Modern Analysis and Topology. Graduate Texts in Mathematics. New York: Springer-Verlag Science & Business Media. ISBN 978-0-387-97986-1. OCLC 31969970. OL 1272666M. <a href="https://search.worldcat.org/oclc/31969970" target="_blank">https://search.worldcat.org/oclc/31969970</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Kelley 1955, p. 163 - Kelley, John (1955). General topology. Graduate Texts in Mathematics. Vol. 27. Springer-Verlag. <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Bourbaki 2007, § 10.2. Theorem 1, Corollary 1. - Bourbaki, Nicolas (2007). Topologie générale. Chapitres 1 à 4. Berlin: Springer. doi:10.1007/978-3-540-33982-3. ISBN 978-3-540-33982-3. <a href="https://link.springer.com/book/10.1007/978-3-540-33982-3" target="_blank">https://link.springer.com/book/10.1007/978-3-540-33982-3</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Mack 1967. - Mack, John (1967). "Directed covers and paracompact spaces". Canadian Journal of Mathematics. 19: 649–654. doi:10.4153/CJM-1967-059-0. MR 0211382. <a href="https://doi.org/10.4153%2FCJM-1967-059-0" target="_blank">https://doi.org/10.4153%2FCJM-1967-059-0</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Bourbaki 2007, § 9.1. Definition 1. - Bourbaki, Nicolas (2007). Topologie générale. Chapitres 1 à 4. Berlin: Springer. doi:10.1007/978-3-540-33982-3. ISBN 978-3-540-33982-3. <a href="https://link.springer.com/book/10.1007/978-3-540-33982-3" target="_blank">https://link.springer.com/book/10.1007/978-3-540-33982-3</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Arkhangel'skii & Fedorchuk 1990, Theorem 5.3.7 - Arkhangel'skii, A.V.; Fedorchuk, V.V. (1990). "The basic concepts and constructions of general topology". In Arkhangel'skii, A.V.; Pontrjagin, L.S. (eds.). General Topology I. Encyclopedia of the Mathematical Sciences. Vol. 17. Springer. ISBN 978-0-387-18178-3. <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Willard 1970 Theorem 30.7. - Willard, Stephen (1970). General Topology. Dover publications. ISBN 0-486-43479-6. <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Gillman & Jerison 1976, §5.6 - Gillman, Leonard; Jerison, Meyer (1976). Rings of continuous functions. Springer-Verlag. <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Robinson 1996, Theorem 4.1.13 - Robinson, Abraham (1996). Non-standard analysis. Princeton University Press. ISBN 978-0-691-04490-3. MR 0205854. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0205854" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0205854</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Arkhangel'skii & Fedorchuk 1990, Theorem 5.2.3 - Arkhangel'skii, A.V.; Fedorchuk, V.V. (1990). "The basic concepts and constructions of general topology". In Arkhangel'skii, A.V.; Pontrjagin, L.S. (eds.). General Topology I. Encyclopedia of the Mathematical Sciences. Vol. 17. Springer. ISBN 978-0-387-18178-3. <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Arkhangel'skii & Fedorchuk 1990, Theorem 5.2.2 - Arkhangel'skii, A.V.; Fedorchuk, V.V. (1990). "The basic concepts and constructions of general topology". In Arkhangel'skii, A.V.; Pontrjagin, L.S. (eds.). General Topology I. Encyclopedia of the Mathematical Sciences. Vol. 17. Springer. ISBN 978-0-387-18178-3. <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Let X = {a, b} ∪ N {\displaystyle \mathbb {N} } , U = {a} ∪ N {\displaystyle \mathbb {N} } , and V = {b} ∪ N {\displaystyle \mathbb {N} } . Endow X with the topology generated by the following basic open sets: every subset of N {\displaystyle \mathbb {N} } is open; the only open sets containing a are X and U; and the only open sets containing b are X and V. Then U and V are both compact subsets but their intersection, which is N {\displaystyle \mathbb {N} } , is not compact. Note that both U and V are compact open subsets, neither one of which is closed. <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Let X = {a, b} and endow X with the topology {X, ∅, {a}}. Then {a} is a compact set but it is not closed. <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>Let X be the set of non-negative integers. We endow X with the particular point topology by defining a subset U ⊆ X to be open if and only if 0 ∈ U. Then S := {0} is compact, the closure of S is all of X, but X is not compact since the collection of open subsets {{0, x} : x ∈ X} does not have a finite subcover. <a href="/wiki/Particular_point_topology" target="_blank">/wiki/Particular_point_topology</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>Arkhangel'skii & Fedorchuk 1990, Corollary 5.2.1 - Arkhangel'skii, A.V.; Fedorchuk, V.V. (1990). "The basic concepts and constructions of general topology". In Arkhangel'skii, A.V.; Pontrjagin, L.S. (eds.). General Topology I. Encyclopedia of the Mathematical Sciences. Vol. 17. Springer. ISBN 978-0-387-18178-3. <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>Steen & Seebach 1995, p. 67 - Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover Publications reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 0507446. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0507446" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0507446</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> </ol>