Weight of Discrete Topology equals Cardinality of Space

Theorem
Let $T = \struct {S, \tau}$ be a discrete topological space.

Then:
 * $\map w T = \size S$

where:
 * $\map w T$ denotes the weight of $T$
 * $\card S$ denotes the cardinality of $S$.

Proof
By Basis for Discrete Topology the set $\BB = \set {\set x: x \in S}$ is a basis of $T$.

By Set of Singletons is Smallest Basis of Discrete Space $\BB$ is smallest basis of $T$:
 * for every basis $\CC$ of $T$, $\BB \subseteq \CC$.

Then by Subset implies Cardinal Inequality:
 * for every basis $\CC$ of $T$, $\card \BB \le \card \CC$.

Hence $\card \BB$ is minimal cardinalty of basis of $T$:
 * $\map w T = \card \BB$ by definition of weight.

Thus by Cardinality of Set of Singletons:
 * $\map w T = \card S$