T4 Space is Preserved under Homeomorphism

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Theorem

Let $T_A = \left({S_A, \tau_A}\right)$ and $T_B = \left({S_B, \tau_B}\right)$ be topological spaces.

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


If $T_A$ is a $T_4$ space, then so is $T_B$.


Proof

Suppose that $T_A$ is a $T_4$ space.


Let $B_1$ and $B_2$ be closed in $T_B$ that are disjoint.

Denote by $A_1 := \phi^{-1} \left({B_1}\right)$ and $A_2 := \phi^{-1} \left({B_2}\right)$ the preimages of $B_1$ and $B_2$ under $\phi$, respectively.


From Preimage of Intersection under Mapping it follows that $A_1$ and $A_2$ are disjoint.

Also, as $\phi$ is a homeomorphism, it is a fortiori continuous.

Thus Continuity Defined from Closed Sets applies to yield that both $A_1$ and $A_2$ are closed.


Now as $T_A$ is a $T_4$ space, we find disjoint open sets $U_1$ containing $A_1$ and $U_2$ containing $A_2$.

From Image of Subset is Subset of Image, we have $A_1 = \phi \left({\phi^{-1} \left({A_1}\right)}\right) \subseteq \phi \left({U_1}\right)$.

Here, the first equality follows from Subset equals Image of Preimage iff Mapping is Surjection, as $\phi$ is a fortiori surjective, being a homeomorphism.

Mutatis mutandis, we deduce also $A_2 \subseteq \phi \left({U_2}\right)$.


From Image of Intersection under Injection it follows that $\phi \left({U_1}\right)$ and $\phi \left({U_2}\right)$ are disjoint.

Since $\phi$ is a homeomorphism, they are also both open in $T_B$.


Therewith, we have construed two disjoint opens in $T_B$, one containing $A_1$, and the other containing $A_2$.

Hence $T_B$ is shown to be a $T_4$ space as well.

$\blacksquare$


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