Simple Loop in Hausdorff Space is Homeomorphic to Quotient Space of Interval

Theorem
Let $\struct { X, \tau_X }$ be a Hausdorff space.

Let $\gamma : \closedint 0 1 \to X$ be a simple loop.

Let $\sim$ be an equivalence relation on $\closedint 0 1$ defined by:

Let $q : \closedint 0 1 \to \closedint 0 1 / \sim$ be the canonical surjection induced by $\sim$.

Let $\tau_q$ be the quotient topology on the quotient space $\closedint 0 1 / \sim$ by $q$.

Let $\tau_\gamma$ be the subspace topology on $\Img \gamma$ induced by $\tau_X$, where $\Img \gamma$ denotes the image of $\gamma$.

Then $\struct {\Img \gamma, \tau_\gamma}$ is homeomorphic to $\struct { \closedint 0 1 / \sim , \tau_q}$.

Proof
Define $\tilde \gamma : \closedint 0 1 \to \Img \gamma$ as the restriction of $\gamma$ to $\closedint 0 1 \times \Img \gamma$.

Restriction of Mapping to Image is Surjection shows that $\tilde \gamma$ is surjective.

Subspace of Hausdorff Space is Hausdorff shows that $\struct {\Img \gamma, \tau_\gamma}$ is a Hausdorff space.

Closed Real Interval is Compact shows that $\closedint 0 1$ is compact.

Continuous Mapping from Compact Space to Hausdorff Space is Closed Mapping shows that $\tilde \gamma$ is a closed mapping.

Continuous Closed Surjective Mapping is Quotient Mapping shows that $\tilde \gamma$ is a quotient mapping.

By definition of simple loop, the equivalence relation $\sim$ is the equivalence induced by $\tilde \gamma$.

Continuous Surjection Induces Continuous Bijection from Quotient Space/Corollary 1 shows that there exists a homeomorphism $f: \closedint 0 1 / \sim \to \Img \gamma$.


 * $\begin{xy} \xymatrix@L+2mu@+1em{

\closedint 0 1 \ar[r]^*{q} \ar[rd]_*{\tilde \gamma} & \closedint 0 1 / \sim \ar@{-->}[d]^*{f} \\ & \Img \gamma }\end{xy}$