Space is Separable iff Density not greater than Aleph Zero
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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
- there exists a countable subset of $T$ which is dense by definition of separable space
- there exists a subset $A$ of $T$ such that $A$ is dense and exists an injection $A \to \N$ by definition of countable set
- there exists a subset $A$ of $T$ such that $A$ is dense and $\card A \le \card \N$ by Injection iff Cardinal Inequality
- 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
- $\map d T \le \aleph_0$ by definition of density
where $\card A$ denotes the cardinality of $A$.
$\blacksquare$
Sources
- Mizar article TOPGEN_1:def 13