View source for Second axiom of countability

{{MSC|54A}}
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A concept in set-theoretic topology: cf. [[Topology, general]]. Topological spaces satisfy the second axiom of countability if they possess a countable [[base]]. The class of spaces which satisfy this axiom was distinguished by F. Hausdorff. This class contains all separable metric spaces (cf. [[Separable space]]). All regular spaces (cf. [[Regular space]]) which satisfy the second axiom of countability are topologically contained in the [[Hilbert cube]], and are thus metrizable and separable (P.S. Urysohn). The study of regular spaces which satisfy the axiom thus leads to the consideration of more concrete objects — subspaces of the Hilbert cube which, by virtue of this fact, acquire a definite topological importance. Finite-dimensional spaces with a countable base allow an even more advanced concretization.



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====References====
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. van Mill,   "Infinite-dimensional topology, prerequisites and introduction" , North-Holland  (1989)  pp. 40</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  pp. 43, 124  (Translated from Russian)</TD></TR>
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