T0 Space is Preserved under Homeomorphism

Theorem
Let $T_A = \left({X_A, \vartheta_A}\right), T_B = \left({X_B, \vartheta_B}\right)$ be topological spaces.

Let $\phi: T_A \to T_B$ be a homeomorphism.

If $T_A$ is a $T_0$ (Kolmogorov) space, then so is $T_B$.

Proof
Let $T_A$ be a $T_0$ (Kolmogorov) space.

By definition:


 * $\forall x, y \in X_A$, either:
 * $\exists U \in \vartheta_A: x \in U, y \notin U$
 * or:
 * $\exists U \in \vartheta_A: y \in U, x \notin U$

Suppose WLOG that:
 * $\exists U \in \vartheta_A: x \in U, y \notin U$

Then:
 * $\left\{{x}\right\} \subseteq U$
 * $\left\{{y}\right\} \subseteq X \setminus U$