Difference between revisions of "Second axiom of countability"
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| − | A concept in set-theoretic topology. Topological spaces satisfy the second axiom of countability if they possess a countable [[ | + | {{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==== | ====References==== | ||
| − | <table><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></table> | + | <table> |
| + | <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> | ||
| + | </table> | ||
Latest revision as of 19:48, 12 December 2015
2020 Mathematics Subject Classification: Primary: 54A [MSN][ZBL]
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.
Comments
References
| [a1] | J. van Mill, "Infinite-dimensional topology, prerequisites and introduction" , North-Holland (1989) pp. 40 |
| [a2] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) pp. 43, 124 (Translated from Russian) |
Second axiom of countability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Second_axiom_of_countability&oldid=36900