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:

$\ds $

$\ds \forall t_1 \in \openint 0 1 , t_2 \in \closedint 0 1 :$

$\ds t_1 \sim t_2 \iff t_2 = t_1$

$\ds $

$\ds \forall t_1 \in \set {0, 1} , t_2 \in \closedint 0 1 :$

$\ds t_1 \sim t_2 \iff t_2 \in \set {0,1}$

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 $\closedint 0 1 / \sim$ induced 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 $\struct { \closedint 0 1 / \sim , \tau_q}$.

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

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

$\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}$

$\blacksquare$