TITLE: All mapping space between CW complexes is a CW complex?
QUESTION [7 upvotes]: Let $\mathrm{Map}(X,Y)$ denote the (unbased) cellular mapping space from $X$ to $Y$.
If $X$ and $Y$ are finite CW complexes, is $\mathrm{Map}(X,Y)$ a CW complex?
Can we know the cell structure of $\mathrm{Map}(X,Y)$?
For example, what is the cell structure of $\mathrm{Map}(S^n,S^k)$ for $n \geq k$?
Please recommend related papers and textbooks.

REPLY [4 votes]: I think I should be posting this as a comment, but as i don't have enough reputation to comment I take the liberty to post this as an answer.  
I am here just elaborating on Eric Wofsey's answer.
Let's take the case, say, $X=\mathbb{N}$ and $Y=I=[0,1]$.  Then 
$Map(X,Y)$ is Hilbert cube.
On the other hand,  the definition of a CW-complex says that a CW complex has to be the union of finite skelta, and 
for reasons explained in Eric Wofsey's answer, this is not the case.