Hausdorff Condition is Preserved under Homeomorphism

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Theorem

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces.

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


If $T_A$ is a $T_2$ (Hausdorff) space, then so is $T_B$.


Proof

By definition of homeomorphism, $\phi$ is a closed continuous bijection.

The result follows from $T_2$ (Hausdorff) Space is Preserved under Closed Bijection.

$\blacksquare$


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