Space is Separable iff Density not greater than Aleph Zero

Theorem
Let $T$ be a topological space.

Then:
 * $T$ is separable $\map d T \le \aleph_0$

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 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 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 $\card A \le \card \N$ 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
 * 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
 * $\map d T \le \aleph_0$ by definition of density

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