Axiomatic foundations of topological spaces

<p>In the <a href="/facts/Mathematics/pxTouaz4">mathematical</a> field of <a href="/facts/Topology/2EqW47cU">topology</a>, a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> is usually defined by declaring its <a href="/facts/Open_set/QfTCIqMu">open sets</a>. However, this is not necessary, as there are many equivalent axiomatic foundations, each leading to exactly the same concept. For instance, a topological space determines a class of <a href="/facts/Closed_set/upYnRTUj">closed sets</a>, of closure and interior operators, and of convergence of various types of objects. Each of these can instead be taken as the primary class of objects, with all of the others (including the class of open sets) directly determined from that new starting point. For example, in <a href="/facts/Kazimierz_Kuratowski/8xoKvo5b">Kazimierz Kuratowski</a>'s well-known textbook on <a href="/facts/Point-set_topology/ezC7B0yP">point-set topology</a>, a topological space is defined as a set together with a certain type of "closure operator," and all other concepts are derived therefrom. Likewise, the neighborhood-based axioms (in the context of <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff spaces</a>) can be retraced to <a href="/facts/Felix_Hausdorff/Rn6DuY2x">Felix Hausdorff</a>'s original definition of a topological space in <a href="/facts/Grundz%25C3%25BCge_der_Mengenlehre/QWK64mH3">Grundzüge der Mengenlehre</a>.
</p><p>Many different textbooks use many different inter-dependences of concepts to develop point-set topology. The result is always the same collection of objects: open sets, closed sets, and so on. For many practical purposes, the question of which foundation is chosen is irrelevant, as long as the meaning and interrelation between objects (many of which are given in this article), which are the same regardless of choice of development, are understood. However, there are cases where it can be useful to have flexibility. For instance, there are various natural notions of <a href="/facts/Convergence_of_measures/fb0I0KAL">convergence of measures</a>, and it is not immediately clear whether they arise from a topological structure or not. Such questions are greatly clarified by the topological axioms based on convergence.
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Axiomatic foundations of topological spaces open-in-new

Axiomatic foundations of topological spaces