Let $X$ be a space which is first countable and compact. Is $X$ necessarily separable? Is $X$ necessarily second countable?
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$[0,1]^\kappa$ is a counterexample if $\kappa>\aleph_0$. – Hanul Jeon Oct 15 '14 at 12:41
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tetori, this space is separable for $\kappa \leqslant 2^{\aleph_0}$. – Tomasz Kania Oct 15 '14 at 12:45
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Is $[0,1]^\kappa$ first countable for $\kappa>\aleph_0$? It would seem that the collection of open sets ${U_\gamma\mid \gamma\in \kappa}$ where $U_\gamma$ is $[0,1/2)$ in the $\gamma$ coordinate and $[0,1]$ elsewhere would be an uncountable family of open sets that wouldn't be contained in any countable local basis at the point 0. – Keith Penrod Oct 15 '14 at 12:54
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This is right. The character of each point in this cube is $\kappa$. – Tomasz Kania Oct 15 '14 at 13:02
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$\pi$-Base, an online database containing information from Steen and Seebach's Counterexamples in Topology, lists the following examples of first countable, compact, non-separable spaces.
Concentric circles
Either-or topology
Lexicographic order topology on the unit square
Uncountable excluded point topology
Steven Clontz
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Paul
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You can learn more about these spaces by viewing the search result. (Paul, thank you for promoting the site.) – Austin Mohr Oct 15 '14 at 23:49
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No, they need not be separable (so not second-countable either). The lexicographic square is a counter-example. Other examples include Alexandrov's duplicates of uncountable compact metric spaces.
Tomasz Kania
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