Characterization of Homeomorphic Topological Spaces/Necessary Condition

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Theorem

Let $T_1 = \struct{S_1, \tau_1}$ be topological space.

Let $S_2$ be a set.

Let $\tau_2$ be a subset of the powerset $\powerset {S_2}$.

Let $\struct{S_2, \tau_2}$ be a topological space homeomorphic to $T_1$.

Then:

there exists a mapping $f : S_1 \to S_2$:

$(1)\quad f$ is a bijection

$(2)\quad f^\to \restriction_{\tau_1}$ is a bijection from $\tau_1$ to $\tau_2$

where

$f^\to \restriction_{\tau_1}$ denotes the restriction of $f^\to$ to $\tau_1$

$f^\to$ denotes the direct image mapping of $f$

Proof

Let $f: S_1 \to S_2$ be a homeomorphism.

By definition of a homeomorphism:

$f$ is a bijection

$f$ is an open mapping

$f$ is a continuous mapping

By definition of an open mapping:

$\forall U \in \tau_1 : f \sqbrk U \in \tau_2$

By definition of direct image mapping:

$\forall U \in \tau_1 : \map {f^\to} U \in \tau_2$

By definition of continuous mapping:

$\forall V \in \tau_2 : f^{-1} \sqbrk V \in \tau_1$

We have:

\(\ds \forall V \in \tau_2: \, \)

\(\ds \map {f^\to} {f^{-1} \sqbrk V}\)

\(=\)

\(\ds f \sqbrk {f^{-1} \sqbrk V}\)

Definition of Direct Image Mapping

\(\ds \)

\(=\)

\(\ds \map {\paren{f \circ f^{-1} } } V\)

Definition of Composite Mapping

\(\ds \)

\(=\)

\(\ds V\)

Image of Preimage of Subset under Surjection equals Subset

It follows that $f^\to \restriction_{\tau_1}$ is a surjection from $\tau_1$ to $\tau_2$.

$\blacksquare$