Space is Separable iff Density not greater than Aleph Zero

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $T$ be a topological space.

Then:

$T$ is separable if and only if $\map d T \le \aleph_0$

where

$\map d T$ denotes the density of $T$,
$\aleph$ denotes the aleph mapping.


Proof

$T$ is separable

if and only if:

there exists a countable subset of $T$ which is dense by definition of separable space

if and only if:

there exists a subset $A$ of $T$ such that $A$ is dense and exists an injection $A \to \N$ by definition of countable set

if and only if:

there exists a subset $A$ of $T$ such that $A$ is dense and $\card A \le \card \N$ by Injection iff Cardinal Inequality

if and only if:

there exists a subset $A$ of $T$ such that $A$ is dense and $\card A \le \aleph_0$ by Aleph Zero equals Cardinality of Naturals

if and only if:

$\map d T \le \aleph_0$ by definition of density

where $\card A$ denotes the cardinality of $A$.

$\blacksquare$


Sources