Are function spaces between CW complexes delta-generated?

Question

Let $X$ and $Y$ be CW complexes. I wish to consider the mapping space $\mathbf{map}(X,Y)$ of continuous maps $X\to Y$, equipped with the (compactly generated version of the) compact-open topology. I believe the path-components and the connected components of this space coincide. (This would be true if $\mathbf{map}(X,Y)$ were a delta-generated or numerically-generated topological space, but I’ve been unable to verify this.) (This came up from reading Lurie’s definition of the homotopy category of spaces in HTT; in order for his definition to be the right one, the path components of $\mathbf{map}(X,Y)$ must coincide with the connected components.)

Answer 1

No, not in general. For instance, consider what happens if $Y=\{0,1\}$ and $X$ is an infinite discrete space. Then the mapping space is just $Y^X$ with the product topology, which is not locally connected and therefore not delta-generated. Or, consider if $Y=[0,1]$ and $X$ is an uncountable discrete space. Then again, the mapping space is $Y^X$ with the product topology, which is not sequential and therefore not delta-generated.

Less trivially, there are in fact examples where the path-components and connected components of $\mathbf{map}(X,Y)$ do not coincide. In particular, consider any example where $X$ is a countable CW complex, $Y$ is connected, and there exists a phantom map $f:X\to Y$, a map whose restriction to each finite subcomplex of $X$ is nullhomotopic but such that $f$ is not nullhomotopic. Write $X$ as an increasing union of finite subcomplexes $K_n$. For each $n$, we can find a map $g_n:X\to Y$ which coincides with $f$ on $K_n$ but which is nullhomotopic. These maps $g_n$ then form a sequence which converges to $f$ in the compact-open topology, and thus also in its $k$-ification (since a convergent sequence together with its limit forms a compact set). So, $f$ is in the closure of the path-component of $\mathbf{map}(X,Y)$ consisting of nullhomotopic maps, but is not in that path-component, so that path-component is not a component.

It is true that $\mathbf{map}(X,Y)$ is delta-generated if $X$ is a finite CW complex. In that case, note first that $\mathbf{map}(X,Y)$ is the colimit of $\mathbf{map}(X,K)$ where $K$ ranges over finite subcomplexes of $Y$ (since every compact subset of $Y$ is contained in a finite subcomplex), so it suffices to consider the case where $Y$ is also finite. Then $Y$ is a retract of an open subset $U\subseteq\mathbb{R}^n$ for some $n$, and the topology of $\mathbf{map}(X,Y)$ is just the topology of uniform convergence with respect to the metric on $Y$ induced from the Euclidean metric. Now for any $f:X\to Y$ and any $g:X\to Y$ sufficiently close to $f$, the linear homotopy from $f$ to $g$ is contained in $U$. Moreover, if $g$ is sufficiently close to $f$, every stage of the composition of this linear homotopy with the retraction $U\to Y$ is close to $f$ (since the retraction is uniformly continuous on any compact set). That is, for any neighborhood $V\subseteq \mathbf{map}(X,Y)$ of $f$, there is a neighborhood $W\subseteq V$ of $f$ such that every element of $W$ is connected to $f$ by a path in $V$. This condition together with first-countability of $\mathbf{map}(X,Y)$ implies it is delta-generated (by a minor modification of the argument for $1\Rightarrow 2$ here).