Let $X$ and $Y$ be CW complexes.

Inside of the space of maps $\mathrm{map}(X,Y)$, we have the subspace $\mathrm{CW}(X,Y)$, consisting of just the cellular maps from $X$ to $Y$. The Cellular Approximation Theorem tells us that the inclusion $\mathrm{CW}(X,Y)\to \mathrm{map}(X,Y)$ induces an isomorphism on $\pi_0$ (in both the pointed and unpointed contexts).

My question: is the inclusion a weak equivalence? (Feel free to use any reasonable topology.)

Let $X$ and $Y$ be CW complexes.

Inside of the space of maps $\mathrm{map}(X,Y)$, we have the subspace $\mathrm{CW}(X,Y)$, consisting of just the cellular maps from $X$ to $Y$. The Cellular Approximation Theorem tells us that the inclusion $\mathrm{CW}(X,Y)\to \mathrm{map}(X,Y)$ induces an isomorphism on $\pi_0$ (in both the pointed and unpointed contexts).

My question: is the inclusion a weak equivalence? (Feel free to use any reasonable topology.)

EDIT: The answer is NO. I'm leaving this here since some interesting ideas are percolating in the comments.