---- > [!definition] Definition. ([[straight-line homotopy]]) > Let $X$ be a [[topological space]] and let $A$ be any [[convex]] [[subspace]] of $\mathbb{R}^{n}$. The **straight-line homotopy** between two [[continuous]] maps $f,g: X \to A$ is the [[homotopy]] $F$ obtained by 'connecting corresponding pairs of points in a line'. Specifically, we define $\begin{align} F & : X \times I \to A \\ F & (x,t):= (1-t)f(x) + tg(x). \end{align}$ > [!justification] > We must show $F$ is a [[homotopy]]. To check [[continuous|continuity]], note that $F(x,t)=F_{1}(x,t)F_{2}(x,t)+F_{3}(x,t)F_{4}(x,t)$ where $\begin{align} F_{1}(x,t):= & 1-t \\ F_{2}(x,t) := & f(x) \\ F_{3}(x,t):= & t \\ F_{4}(x,t) := & g(x), \end{align}$ and is therefore [[continuous]] on $X$ as a sum of products of [[continuous]] functions on $X$. - [ ] picture ---- #### ---- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM outgoing([[]]) FLATTEN file.tags GROUP BY file.tags as Tag