# Topology up to isomorphism

Let $$A$$ be a set, $$\mid A \mid$$=$$\aleph_0$$ (assume that A$$\subseteq\mathbb{R}$$) ,

and let $$\chi$$=$$\{\tau\subseteqP(A)\mid($$A,$$\tau)$$is a Hausdorff space, $$\mid \tau \mid$$=$$\aleph\}$$ .
($$P(A)$$ is the power set of $$A$$)

What is the "value" of $$\mid \chi \mid$$ $$?$$

The question is basically how many Hausdorff topologies $$\tau$$ ($$\mid \tau \mid=\aleph$$) (up to isomorphism) are exist on a set $$A$$, $$\mid A \mid=\aleph_0$$.

Correct me if I'm wrong, but I know that there is at least one topology as I mentioned before:

For example:

The set is $$\mathbb{Q}$$ and the topology $$\tau$$ is Subspace topology of the euclidean metric on $$\mathbb{R}$$ (its Hausdorff).

Obviously $$\mid \mathbb{Q} \mid$$=$$\aleph_0$$, and we know that the base of $$\tau$$ in $$\mathbb{R}$$ is all the open intervals, hence $$\mid \tau \mid$$ on $$\mathbb{Q}$$ is $$\aleph$$.

Hence $$\chi \neq \emptyset$$.

What is $$\mid \chi \mid$$?

• $\aleph$ is not a definite cardinal? What do you mean, just that $\tau$ is infinite ? – Henno Brandsma Aug 22 at 8:40
• @HennoBrandsma I suppose that by $\aleph$ he means the cardinality of the real numbers. – Cronus Aug 22 at 8:44
• @Henno: I think that you should have seen the notation $\aleph$ for the cardinality of the reals already. It's been used quite extensively in tags you seem to be participating in. – Asaf Karagila Aug 22 at 8:55
• The title does not exactly fit the question.: are you interested in the number of topologies up to isomorphism, or just the number of topologies ? (The former question may be harder to answer, although I suspect it's the same number) – Maxime Ramzi Aug 22 at 10:00
• @AsafKaragila I think I’ve seen it before, but it’s old-fashioned. Most of the modern books I have use $\mathfrak{c}$. – Henno Brandsma Aug 22 at 14:32

There are at most $$\kappa:=2^{2^{\aleph_0}}$$ topologies on $$A$$. Also, there are that many non-equivalent free ultrafilters on $$A$$, and each of those gives rise that a mutually non-homeomorphic Hausdorff topology on $$A$$ (by using it as the set of neighbourhoods of a unique non-isolated point). So there are $$\kappa$$ many Hausdorff topologies on $$A$$ (all of size of the reals). The answer is thus the size of the power set of $$\Bbb R$$.