Maximum Cardinality of Separable Hausdorff Space

Theorem

Let $T = \struct {S, \tau}$ be a Hausdorff space which is separable.

Then $S$ can have a cardinality no greater than $2^{2^{\aleph_0} }$.

Proof

Let $D$ be an everywhere dense subset of $S$ which is countable, as is guaranteed as $T$ is separable.

Consider the mapping $\Phi: S \to 2^{\powerset D}$ defined as:

$\forall x \in S: \map {\map \Phi x} A = 1 \iff A = D \cap U_x$ for some neighborhood $U_x$ of $x$

This article, or a section of it, needs explaining. In particular: It is not clear in Steen & Seeabch what is meant by $\Phi: S \to 2^{\powerset D}$ -- presumably $2^{\powerset D}$ is the ordinal which is the power set of the power set of $D$. It is also not clear what the notation $\map {\map \Phi x} A$ means -- in fact is may be the case that a transcription error has been committed. Hence the proof cannot be attempted until these points have been cleared up. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

It is seen that if $T$ is a Hausdorff space, then $\Phi$ is an injection.

It follows that:

$\card S \le \card {2^{\powerset D} } = 2^{2^{\aleph_0} }$

This article, or a section of it, needs explaining. In particular: the chain of reasoning leading to the above You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Compactness Properties and the $T_i$ Axioms