Simple Connectedness is Preserved under Homeomorphism

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Theorem

Let $\struct {T_1, \tau_1}, \struct {T_2, \tau_2}$ be topological spaces.

Let $\phi: T_1 \to T_2$ be a homeomorphism.

Let $S \subseteq T_1$ be a subset of $T_1$.

Let $S$ be simply connected in $\struct {T_1, \tau_1}$.


Then $\phi \sqbrk S$ is simply connected in $\struct {T_2, \tau_2}$.

That is, simple connectedness is a topological property.


Proof

By Definition of Simply Connected, $S$ is path-connected.

Path-Connectedness is Preserved under Homeomorphism shows that $\phi \sqbrk S$ is path-connected.

Let $x \in S$.

Let:

$\gamma_1 : \closedint 0 1 \to \phi \sqbrk S$
$\gamma_2 : \closedint 0 1 \to \phi \sqbrk S$

be two loops in $\phi \sqbrk S$.

Let $\gamma_1, \gamma_2$ both have base point $x$.

Composite of Continuous Mappings is Continuous shows that $\phi^{-1} \circ \gamma_1$ and $\phi^{-1} \circ \gamma_2$ are continuous mappings.

So $\phi^{-1} \circ \gamma_1 , \phi^{-1} \circ \gamma_2$ are loops in $S$ with base point $\map {\phi^{-1} } x$.


By Definition of Simply Connected, there exists a path homotopy $H: \closedint 0 1 \times \closedint 0 1 \to S$ such that:

\(\ds \forall s \in \closedint 0 1: \, \) \(\ds \map H {s, 0}\) \(=\) \(\ds \map {\phi^{-1} \circ \gamma_1} s\)
\(\ds \forall s \in \closedint 0 1: \, \) \(\ds \map H {s, 1}\) \(=\) \(\ds \map {\phi^{-1} \circ \gamma_2} s\)
\(\ds \forall t \in \closedint 0 1: \, \) \(\ds \map H {0, t}\) \(=\) \(\ds \map {\phi^{-1} \circ \gamma_1} 0\) \(\ds = \map {\phi^{-1} } x\)
\(\ds \forall t \in \closedint 0 1: \, \) \(\ds \map H {1, t}\) \(=\) \(\ds \map {\phi^{-1} \circ \gamma_1} 1\) \(\ds = \map {\phi^{-1} } x\)


Composite of Continuous Mappings is Continuous shows that $ \phi \circ H : \closedint 0 1 \times \closedint 0 1 \to \phi \sqbrk S$ is a continuous mapping.

Then:

\(\ds \forall s \in \closedint 0 1: \, \) \(\ds \map {\phi \circ H} {s, 0}\) \(=\) \(\ds \map {\phi \circ \phi^{-1} \circ \gamma_1} s\) \(\ds = \map {\gamma_1} s\)
\(\ds \forall s \in \closedint 0 1: \, \) \(\ds \map {\phi \circ H} {s, 1}\) \(=\) \(\ds \map {\phi \circ \phi^{-1} \circ \gamma_2} s\) \(\ds = \map {\gamma_2} s\)
\(\ds \forall t \in \closedint 0 1: \, \) \(\ds \map {\phi \circ H} {0, t}\) \(=\) \(\ds \map {\phi \circ \phi^{-1} \circ \gamma_1} 0\) \(\ds = x\)
\(\ds \forall t \in \closedint 0 1: \, \) \(\ds \map {\phi \circ H} {1, t}\) \(=\) \(\ds \map {\phi \circ \phi^{-1} \circ \gamma_1} 1\) \(\ds = x\)

It follows that $\phi \circ H$ is a path homotopy between $\gamma_1$ and $\gamma_2$.

By Definition of Simply Connected, $\phi \sqbrk S$ is simply connected.

$\blacksquare$


Sources

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