Countable But Not Second Countable Topological Space

[Nolen Ryba]

I'm wondering if someone can furnish me with either an example of a topological space that is countable (cardinality) but not second countable or a proof that countable implies second countable. Thanks.

 

[morphism]

Take the countable set $\mathbb{N}\times\mathbb{N}$. Topologize it by making any set that doesn't contain (0,0) open, and if a set does contain (0,0), it's open iff it contains all but a finite number of points in all but a finite number of columns. (Draw a picture. If an open set contains (0,0), then it can only miss infinitely many points in a finite number of columns, while it misses finitely many points in all the other columns.)

Now this topology doesn't have a countable base at (0,0), so it's not first countable let alone second countable.

Source: Steen & Seebach, Counterexamples in Topology, page 54. They call it the Arens-Fort Space.

 

[Nolen Ryba]

Thanks! I own that book so I'll be having a look pretty soon.

 

[morphism]

I've been thinking about this a little bit more, and I believe I have another example, although it's a bit more 'sophisticated' in that it requires a bit of advanced set theory.

This time our countable set is X = $\mathbb{N} \cup \{\mathbb{N}\}$. Take any free ultrafilter F on $\mathbb{N}$, and define a topology on X by letting each subset {n} of $\mathbb{N}$ be open, and defining nbhds of $\left{\mathbb{N}\right}$ to be those of the form $\{\mathbb{N}\} \cup U$, where U is in F. As in the previous example, this topology fails to have a countable base at $\left{\mathbb{N}\right}$ (because we cannot have a countable base for any free ultrafilter on the naturals), so again it fails to be first countable.

Edit:
Hmm... Now I'm wondering if there's a countable space that's first countable but not second countable!

Edit2:
Maybe that was silly. If X is countable and has a first countable topology, then the union of the bases at each of its points is the countable union of countable sets and is hence countable (and a basis for the topology). So, I'm lead to conclude that a countable, first countable space is necessarily second countable.