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All of the examples of non-first countable spaces I have seen are uncountable (for instance any uncountable set with the cofinite topology). I would like to know if every countably infinite $T_1$ space $X$ is first countable. Since $\{A\subseteq X|x\in A\}$ for a given $x\in X$ is uncountable, there doesn't seem to be a 1-line proof; a proof should require use of the axioms of a topology.

Perhaps I haven't seen a proof or counterexample anywhere because I haven't looked in the right place or am simply missing an "obvious" proof or counterexample. Maybe someone can point me to a reference or exercise in a textbook where this shows up.

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The best place to look for things like this is Steen and Seebach, Counterexamples in Topology. – Nate Eldredge Apr 19 '12 at 12:56
up vote 10 down vote accepted

Look up Arens-Fort space.

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Very cool, thanks! – J.K.T. Apr 19 '12 at 12:56

(What follows is adapted from a couple of old sci.math posts of mine, from 2001 and 2008. URLs below if anyone is interested.)

It is possible for countable space, even a countable regular Hausdorff space, to not be first countable. The key in making this happen is that although each subset of a countable space must be countable, a collection of subsets of a countable space can be uncountable. In fact, there even exist countable (regular Hausdorff) spaces that have no points of first countability, where $x \in X$ is a point of first countability of the topological space $X$ means that every neighborhood of $x$ (when viewed as a topological space with the subspace topology inherited from $X$) fails to be first countable. For some examples, see:

Peter Wamer Harley, A countable nowhere first countable Hausdorff space, Canadian Mathematical Bulletin 16 (1973), 441-442. [.pdf file of Harley's paper]

Ronald [Ronnie] Fred Levy, Countable spaces without points of first countability, Pacific Journal of Mathematics 70 (1977), 391-399. [Proposition 2.1 gives $2^c$ many pairwise non-homeomorphic countable regular Hausdorff spaces, each of which has no points of first countability.] [.pdf file of Levy's paper]

Richard Curtis Willmott, Countable yet nowhere first countable, Mathematics Magazine 52 (1979), 26-27.

Besides Levy, Leslie Owen Foged also constructed $2^c$ nonhomeomorphic countable spaces having no points of first countability in his 1979 Ph.D. Dissertation (under Ron Freiwald, Washington University) Weak Bases for Topological Spaces. I believe the spaces Foged constructed were also Hausdorff and regular, but I'm not certain about this.

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Thanks Dave. I am curious to know if any of these countable, nowhere first countable spaces are homogeneous. – J.K.T. Apr 19 '12 at 20:53
I see. Yes, Levy's examples are homogeneous. – J.K.T. Apr 19 '12 at 21:04
@J.T.: The $2^c$ many examples Levy gives are all homogeneous, but I don't know about the others. After submitting this I saw your second comment, but don't know how to delete my comment. – Dave L. Renfro Apr 19 '12 at 21:09

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