Cardinality of Set is Topological Property

Theorem

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

The cardinality of $S$ is a topological property of $T$.

Proof

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.

Let $T_1$ and $T_2$ be homeomorphic.

Then by definition there exists a homeomorphism $f: T_1 \to T_2$.

Hence by definition $f$ is a bijection.

Hence by definition $S$ and $T$ are equjivalent.

That is, they have the same cardinality.

Thus cardinality is preserved by homeomorphism.

Hence the result by definition of topological property.

$\blacksquare$

Examples

Set with $7$ Elements

Let $P$ be the property defined as:

$\map P M := $ The set $M$ contains $7$ elements.

Then $P$ is a topological property.