Cardinality of Set is Topological Property
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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:
Then $P$ is a topological property.