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:

$\map w T = \card S$

$\blacksquare$

Sources

• 1989: Ryszard Engelking: General Topology (revised and completed ed.)

• Mizar article TOPGEN_2:14