Second-countable space

<h2 id="properties">Properties</h2>
<p>Second-countability is a stronger notion than <a href="/facts/First-countable_space/HDuTYa6A">first-countability</a>. A space is first-countable if each point has a countable <a href="/facts/Local_base/fMRM8Xht">local base</a>. Given a base for a topology and a point <i>x</i>, the set of all basis sets containing <i>x</i> forms a local base at <i>x</i>. Thus, if one has a countable base for a topology then one has a countable local base at every point, and hence every second-countable space is also a first-countable space. However any uncountable <a href="/facts/Discrete_space/hpvXA23u">discrete space</a> is first-countable but not second-countable.
</p><p>Second-countability implies certain other topological properties. Specifically, every second-countable space is <a href="/facts/Separable_space/2bmU73bk">separable</a> (has a countable <a href="/facts/Dense_(topology)/nPT9Yxao">dense</a> subset) and <a href="/facts/Lindel%25C3%25B6f_space/HC4t6bvz">Lindelöf</a> (every <a href="/facts/Open_cover/2VUdelA4">open cover</a> has a countable subcover). The reverse implications do not hold. For example, the <a href="/facts/Lower_limit_topology/uE1nTKxG">lower limit topology</a> on the real line is first-countable, separable, and Lindelöf, but not second-countable. For <a href="/facts/Metric_space/gnBBRXtc">metric spaces</a>, however, the properties of being second-countable, separable, and Lindelöf are all equivalent.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>  Therefore, the lower limit topology on the real line is not metrizable.
</p><p>In second-countable spaces—as in metric spaces—<a href="/facts/Compact_space/d0cgXJH7">compactness</a>, sequential compactness, and countable compactness are all equivalent properties.
</p><p><a href="/facts/Urysohn%2527s_metrization_theorem/BCLdo5Jf">Urysohn's metrization theorem</a> states that every second-countable, <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff</a> <a href="/facts/Regular_space/gLvRIhRH">regular space</a> is <a href="/facts/Metrizable/BCLdo5Jf">metrizable</a>. It follows that every such space is <a href="/facts/Completely_normal_space/t4DLqLCd">completely normal</a> as well as <a href="/facts/Paracompact/rjTrvF7j">paracompact</a>. Second-countability is therefore a rather restrictive property on a topological space, requiring only a separation axiom to imply metrizability.
</p>
<h3>Other properties</h3>
<ul><li>A continuous, <a href="/facts/Open_map/abHUKX7l">open</a> <a href="/facts/Image_(mathematics)/KANefMAl">image</a> of a second-countable space is second-countable.</li>
<li>Every <a href="/facts/Subspace_(topology)/NjU6z4Ca">subspace</a> of a second-countable space is second-countable.</li>
<li><a href="/facts/Quotient_space_(topology)/muE0qQVC">Quotients</a> of second-countable spaces need not be second-countable; however, <i>open</i> quotients always are.</li>
<li>Any countable <a href="/facts/Product_space/0hg02IaX">product</a> of a second-countable space is second-countable, although uncountable products need not be.</li>
<li>The topology of a second-countable T1 space has <a href="/facts/Cardinality/m6VPojwT">cardinality</a> less than or equal to <i>c</i> (the <a href="/facts/Cardinality_of_the_continuum/ob578Q9l">cardinality of the continuum</a>).</li>
<li>Any base for a second-countable space has a countable subfamily which is still a base.</li>
<li>Every collection of disjoint open sets in a second-countable space is countable.</li></ul>
<h2 id="examples">Examples</h2>
<ul><li>Consider the disjoint countable union 
  
    
      
        X
        =
        [
        0
        ,
        1
        ]
        ∪
        [
        2
        ,
        3
        ]
        ∪
        [
        4
        ,
        5
        ]
        ∪
        ⋯
        ∪
        [
        2
        k
        ,
        2
        k
        +
        1
        ]
        ∪
        ⋯
      
    
    {\displaystyle X=[0,1]\cup [2,3]\cup [4,5]\cup \dots \cup [2k,2k+1]\cup \dotsb }
  
. Define an equivalence relation and a <a href="/facts/Quotient_topology/muE0qQVC">quotient topology</a> by identifying the left ends of the intervals - that is, identify 0 ~ 2 ~ 4 ~ … ~ 2k and so on. <i>X</i> is second-countable, as a countable union of second-countable spaces. However, <i>X</i>/~ is not first-countable at the coset of the identified points and hence also not second-countable.</li>
<li>The above space is not homeomorphic to the same set of equivalence classes endowed with the obvious metric: i.e. regular Euclidean distance for two points in the same interval, and the sum of the distances to the left hand point for points not in the same interval -- yielding a strictly coarser topology than the above space.  It is a separable metric space (consider the set of rational points), and hence is second-countable.</li>
<li>The <a href="/facts/Long_line_(topology)/bgkMDrpB">long line</a> is not second-countable, but is first-countable.</li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Stephen Willard, <i>General Topology</i>, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.</li>
<li>John G. Hocking and Gail S. Young (1961). <i>Topology.</i> Corrected reprint, Dover, 1988.  <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-65676-4</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Willard, theorem 16.11, p. 112 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Second-countable space open-in-new

Second-countable space