Proposition

Homotopy equivalence is an equivalence relation on the class of all topological spaces.

Solution

Let $X, Y, Z$ be given.

The identity mapping $i_X$ on $X$ is continuous. Moreover, $i_X \circ i_X = i_X$, so $i_X \circ i_X \simeq i_X$. Therefore, $X \simeq X$.

Suppose $X \simeq Y$. Then there exist continuous functions $\phi: X \rightarrow Y, \psi: Y \rightarrow X$ such that $\phi \circ \psi \simeq i_Y$ and $\psi \circ \phi \simeq i_X$. This means we have $\psi$ and its homotopy inverse, so $Y$ is homotopic equivalent to $X$.

Suppose $X \simeq Y$ and $Y \simeq Z$. Then there exist continuous functions $\phi: X \rightarrow Y, \phi’: Y \rightarrow Z$ and their homotopy inverses $\psi: Y \rightarrow X, \psi’: Z \rightarrow Y$, respectively.

$\phi’ \circ \phi$ is a continuous map from $X$ to $Z$ and we claim that $\psi \circ \psi’$ is a homotopy inverse for $\phi’ \circ \phi$. Let $F, F’$ be homotopies connecting $\phi \circ \psi$ to $i_Y$, and $\phi’ \circ \psi’$ to $i_Z$, respectively.

Let $G: Z \times I \rightarrow Z$ such that

\[\begin{align*} G(z, t) &= \begin{cases} \phi'(F(\psi'(z), 2t)) & (t \in [0, 1/2]) \\ F'(z, 2t - 1) & (t \in [1/2, 1]). \end{cases} \end{align*}\]

$G$ is continuous by the pasting lemma.

  • $G(z, 0) = \phi’(F(\psi’(z), 0)) = \phi’(\phi(\psi(\psi’(z))))$.
  • $G(z, 1) = F’(z, 1) = i_Z(z) = z$.

Thus $G$ is a homotopy connecting $\phi’ \circ \phi \circ \psi \circ \psi’$ to $i_Z$.

Now let $F, F’$ be homotopies connecting $\psi \circ \phi$ to $i_X$, and $\psi’ \circ \phi’$ to $i_Y$, respectively. (Overriding the notations since I’m running out of suitable letters.)

Similarly, let $H: X \times I \rightarrow X$ such that

\[\begin{align*} H(x, t) &= \begin{cases} \psi(F'(\phi(x), 2t)) & (t \in [0, 1/2]) \\ F(x, 2t - 1) & (t \in [1/2, 1]). \end{cases} \end{align*}\]

$H$ is continuous by the pasting lemma.

  • $G(z, 0) = \phi’(F(\psi’(z), 0)) = \phi’(\phi(\psi(\psi’(z))))$.
  • $G(z, 1) = F’(z, 1) = i_Z(z) = z$.

Thus $H$ is a homotopy connecting $\psi \circ \psi’ \circ \phi’ \circ \phi$ to $i_X$.

Therefore, $\psi \circ \psi’$ is a homotopy inverse of $\phi’ \circ \phi$, so $X \simeq Z$.