Looking for examples of first countable, compact spaces which is not separable

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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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

$\pi$-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. You can learn more about these spaces by viewing the search result.

Concentric circles

Either-or topology

Lexicographic order topology on the unit square

Uncountable excluded point topology

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