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Could someone give me some classical examples of first countable, compact spaces which is not separable?

However, other examples are also welcome. Any help will be appreciated.

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3 Answers

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The easiest way to get Hausdorff examples of non-separable, compact, first countable spaces is exemplified by the lexicographically ordered square that Austin has already mentioned. Let $X$ be an uncountable compact first countable linearly ordered space with the order topology; the closed unit interval and the Cantor set are obvious examples. Let $Y$ be any compact first countable linearly ordered space with more than two points. Endow the product $X\times Y$ with the order topology induced by the lexicographic order: $\langle x,y\rangle \prec \langle x',y'\rangle$ iff either $x<x'$, or $x=x'$ and $y<y'$. When $X=Y=[0,1]$ you get the lexicographically ordered square.

If you’re willing to consider consistency results, you can have a compact, connected, hereditarily Lindelöf, first countable, non-separable, perfectly normal, linearly ordered space in which every collection of pairwise disjoint, non-empty open sets is countable (i.e., a Suslin continuum): such a space can be constructed using the combinatorial principle $\diamondsuit$, which is consistent with ZFC (and implies CH).

Of course any product of countably many non-separable, first countable, compact Hausdorff spaces is also such a space. It’s probably worth noting that all such spaces have cardinality $2^\omega$, since every first countable, compact Hausdorff space is either countable (hence separable) or of cardinality $2^\omega$. In fact, results of A. Gryzlov in Two theorems on the cardinality of topological spaces, Soviet Math. Dokl.21(1980), 506-509, and V. I. Malyhin, On the power of first countable $T_1$-bicompacta, Colloq. Math. Soc. Janos Bolyai, 1978, 827-828, show that this is true even for $T_1$-spaces. However, I don’t offhand know of any interesting $T_1$ examples that aren’t Hausdorff.

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A general construction for such spaces: copy the idea of the Alexandrov double.

Take a compact first countable (Hausdorff) space $X$ and let the space $S$ be the disjoint union of 2 copies of $X$, say $X_1$ and $X_2$ (where for $x$ in $X$ we denote by $x_1$ its counterpart in $X_1$, $x_2$ its counterpart in $X_2$, and similarly for $A_1$, $A_2$ when $A \subset X$) with the topology that all points of $X_2$ are isolated points, while a basic open neighborhood for $x_1$ in $X_1$ is of the form $U_1 \cup U_2\setminus \{x_2\}$, where $U$ is an open neighborhood of $x$ in $X$.

One checks that $X_1$ as a subspace of $S$ is still homeomorphic to $X$, $X_2$ is an open and discrete subspace of $S$ and $S$ is first countable iff $X$ is, and $S$ is also compact Hausdorff iff $X$ is.

The fact that $X_2$ is open and discrete implies that every dense subset of $S$ must contain all of $X_2$, so for e.g. $[0,1]$ or the circle we get examples of spaces like you asked for: non-separable, first countable, compact (Hausdorff).

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Wow, that's a nice idea of constructing the example! –  Paul Jan 13 '12 at 23:48
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