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Consider the set of topological morphisms $\mathrm{mor_{TOP}}(X,Y)$.

By definition, the topological morphisms $f,\,g: X \longrightarrow Y$ are homotopic, if $$ \exists\; (H: X \times I \longrightarrow Y) \left( H_0 = f \land H_1=g \right)\,. $$

Is there a nontrivial topology on the set of morphisms $\mathrm{mor_{TOP}}(X,Y)$, such that $$ \begin{align} \phi(H):I & \longrightarrow \mathrm{mor_{TOP}}(X,Y) \\ t & \mapsto H_t \end{align} $$ is continuous, and such that this defines a well-defined map of sets $$ \phi:\mathrm{mor_{TOP}}(X \times I,Y) \longrightarrow % \mathrm{mor_{TOP}}(X,Y) $$ Maybe even one that is not just artificially constructed, but comes natural in a certain way (and possible, such that $\phi$ is continuous)?

Somehow (possibly because a professor once made a remark) this vague idea, that a homotopy between continuous maps is something like path, is stuck in my head.

I'm aware that paths can be viewed as homotopies between points (viewed as constant maps with the one-point space as their domain). But I'm interested in this 'other direction'.

I also quickly glanced upon the compact-open topology article on Wikipedia (the only general topological construction for the morphism sets I heard of), but I didn't find the word 'homotopy' in the article's 'text body', so I suppose it doesn't give an answer to my question.

UPDATE: I've only just now found this question and the one referred to in the comment (on MO).
I still do think my question to be different: I don't need a topology that would induce a correspondence or anything like that, just what I wrote here would be interesting enough.

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I noticed you found out references to answer your question, but let me write up something to get it off the unanswered queue.

Let $X, Y, Z$ be locally compact Hausdorff spaces. Then the map $C(X \times Y, Z) \to C(X, C(Y, Z))$ given by sending $f(-, -)$ to $x \mapsto f(x, -)$ is in fact a homeomorphism, where function spaces are imposed the compact-open topology.

It is a fact that if $P, Q, R$ are locally compact Hausdorff spaces, then deciding continuity of a map $f : P \times Q \to R$ is equivalent to deciding continuity of the corresponding map $g : P \to C(Q, R)$.

Using this, you can see continuity of $\varphi$ is implied by continuity of the corresponding map $C(X \times Y, Z) \times X \to C(Y, Z)$, which is implied by continuity of the corresponding map $C(X \times Y, Z) \times X \times Y \to Z$, which is basically just the evaluation map $e : C(A, B) \times A \to B$ given by $e(f, a) = f(a)$, for $A = X \times Y, B = Z$. Continuity of $\varphi^{-1}$ is decided analogously.

The point-set topology aside, this means the compact-open topology makes the category of locally compact Hausdorff spaces cartesian closed. This is very useful in general, for example in the interpretation you said: a homotopy $F : I \times X \to Y$ is "the same" as a map $f : I \to C(X, Y)$, which is a path in the space of functions. This implies, say, that $\pi_1(\Omega X) \cong \pi_2 X$ where $\Omega X$ is the "loop space" of $X$, the space of (based) loops in $X$ with compact-open topology.

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