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> [!proposition] Proposition. ([[continuous functions respect path homotopy]])
> Let $X,Y$ be [[topological space|topological spaces]] and $k:X \to Y$ [[continuous]]. Let $H$ be a [[path homotopy]] between [[parameterized curve|paths]] $f,g$ in $X$. Then we quickly have
> 1. $k \circ H$ is a [[path homotopy]] between $k \circ f$ and $k \circ g$;
> 2. $k \circ (f * g)=(k \circ f)* (k \circ g)$,
>
> where $*$ denotes the [[fundamental groupoid|path concatentation]] operation.
> ![[CleanShot 2024-03-28 at
[email protected]]]
> [!proof]- Proof. ([[continuous functions respect path homotopy]])
> The proof is a triviality. Explicitly, let $x_{0},x_{1}$ be initial and final points of $f$ and $g$. Then $k(x_{0}),k(x_{1})$ are initial/final points of $k \circ f$ and $k\circ g$.
> 1. $k \circ F$ is [[continuous]] as a [[composition of continuous functions is continuous|continuous composition]]. And for all $s$, $(k \circ F)(s, 0)=k(F(s,0))=k(f(s))$ and $(k \circ F)(s,1)=k(F(s,1))=k(g(s))$. And $(k \circ F)(0, t)=k(x_{0})$, $(k \circ F)(1,t)=k(x_{1})$.
>
> 2. This is straight from the definition of $*$.
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#### 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