Basic Theorems Regarding Homotopies

Recall from the Homotopic Mappings Relative to a Subset of a Topological Space page that if $X$ and $Y$ are topological spaces and $A \subseteq X$ is a subspace, and $f, g : X \to Y$ are continuous functions then $f$ is said to be homotopic to $g$ relative to $A$ if there exists a continuous function $H : X \times I \to Y$ such that:

• 1) $H_0 = f$.

• 2) $H_1 = g$.

• 3) $H(a, t) = f(a) = g(a)$ for all $a \in A$ and for all $t \in I$

In such cases we write $f \simeq_A g$.

Furthermore, if $A = \emptyset$ then we simply say that $f$ is homotopic to $g$ and write $f \simeq g$.

We will now state some basic theorems regarding homotopies.

Theorem 1: Let $X$ and $Y$ be topological spaces and let $f, g : X \to Y$ be embeddings. If $f$ is isotopic to $g$ then $f$ is homotopic to $g$.

• Proof: Since $f$ is isotopic to $g$ there exists a continuous function $H : X \times I \to Y$ such that $H_t : X \to Y$ is an embedding for each $t \in I$, $H_0 = f$, and $H_1 = g$. So we can completely ignore this first condition to see that $f$ is homotopic to $g$. $\blacksquare$