Homotopy

Remarks

For each $t \in I$, the map $h_t = H(\cdot, t) \colon X \to Y$ is continuous, so a homotopy is a continuous family $(h_t)_{t \in I}$ of maps with $h_0 = f$ and $h_1 = g$. Example: for $X = Y = \R$, $f = \id$ and $g = 0$, the map $H(x, t) = (1 - t)x$ is continuous, with $H(x, 0) = x$ and $H(x, 1) = 0$, so it is a homotopy from $f$ to $g$. Non-example: for the same $f$ and $g$, the map $K(x, t) = x$ for $t < 1$ and $K(x, 1) = 0$ has the right ends but is not continuous at $(1, 1)$, so it is not a homotopy. If $H$ is a homotopy from $f$ to $g$, then $(x, t) \mapsto H(x, 1 - t)$ is a homotopy from $g$ to $f$.