Homeomorphic Topological Spaces Are Homotopically Equivalent
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Recall from the Homotopically Equivalent Topological Spaces page that two topological spaces $X$ and $Y$ are said to be homotopically equivalent if there exists continuous functions $f : X \to Y$ and $g : Y \to X$ such that:

\begin{align} \quad g \circ f = \mathrm{id}_X \end{align}
\begin{align} \quad f \circ g = \mathrm{id}_Y \end{align}

We now prove a simple result which states that any two homeomorphic topological spaces must be homotopically equivalent.

Theorem 1: Let $X$ and $Y$ be topological spaces. If $X$ is homeomorphic to $Y$ then $X$ is homotopically equivalent to $Y$.
  • Proof: Suppose that $X$ is homeomorphic to $Y$. Then there exists a homeomorphism $f : X \to Y$. Observe that the homeomorphism $f$ by definition is bijective, continuous, and $f^{-1}$ is continuous. Let $g : Y \to X$ be defined by $g = f^{-1}$. Then as remarked, $g$ is continuous. Furthermore:
\begin{align} \quad g \circ f = f^{-1} \circ f = \mathrm{id}_X \end{align}
\begin{align} \quad f \circ g = f \circ f^{-1} = \mathrm{id}_Y \end{align}
  • So $X$ is homotopically equivalent to $Y$. $\blacksquare$
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