Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints

Write $I=[0,1]$ with its usual subspace topology, as in Paths, path-connected spaces and path components. Let $X$ and $Y$ be topological spaces, and let $f,g:X\to Y$ be continuous maps (Continuity of a map of topological spaces at a point and globally).

A homotopy from $f$ to $g$ is a continuous map

$H:X\times I\longrightarrow Y$

from the product space (The product set $\prod_{i \in I} X_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space) such that $H(x,0)=f(x)$ and $H(x,1)=g(x)$ for every $x\in X$. When such an $H$ exists, $f$ and $g$ are homotopic, written $f\simeq g$.

Let $A\subseteq X$ carry the subspace topology (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace). The homotopy $H$ is a homotopy relative to $A$, or a homotopy rel $A$, when

$H(a,t)=f(a)=g(a)\qquad(a\in A,\ t\in I).$

Thus a homotopy rel $A$ can exist only when $f|_A=g|_A$, and every ordinary homotopy is a homotopy rel $\varnothing$. We write $f\simeq_A g$ when a homotopy rel $A$ exists.

If $\alpha,\beta:I\to Y$ are paths with the same initial point and the same terminal point (Paths, path-connected spaces and path components), a path homotopy from $\alpha$ to $\beta$ relative to the endpoints is a homotopy $H:I\times I\to Y$ rel $\{0,1\}$. Explicitly,

$H(s,0)=\alpha(s),\quad H(s,1)=\beta(s),\quad H(0,t)=\alpha(0)=\beta(0),\quad H(1,t)=\alpha(1)=\beta(1).$

The first coordinate $s$ parametrises the path and the second coordinate $t$ parametrises the deformation.

• The adjective relative means pointwise fixed throughout the deformation, not merely mapped back into $A$.
• A homotopy is a map on a product. A family of maps $H_t(x):=H(x,t)$ is not by itself a homotopy unless the joint map $(x,t)\mapsto H_t(x)$ is continuous.