Is an image of a cellular map CW-complex?

Question

Let $X$ and $Y$ be two CW-complexes and $f:X\longrightarrow Y$ be a cellular map.

Is $f(X)$ a CW-complex?
If the answer is no, then under what conditions $f(X)$ is a CW-complex?

Thanks in advance.

Answer 1

No, the image of a cellular map need not be a CW-complex. For instance, let $X=Y=S^2$ with the cell structure having one $0$-cell $p$ and one $2$-cell. For a map $f:X\to Y$ to be cellular, then, the only requirement is that $f(p)=p$. The image of such a map need not be a CW-complex. For instance, you could first take any continuous surjection $S^2\to [0,1]$ and then compose it with a path $[0,1]\to S^2$ that draws a Hawaiian earring.

I don't know of any particularly natural condition on maps between arbitrary CW-complexes that will guarantee the image is a CW-complex (other than just directly demanding that the image of each cell is a union of cells). If you instead have something stronger like a simplicial complex (or the geometric realization of a simplicial set) then you could ask for $f$ to be a simplicial map (in which case the image is not just a CW-complex but a subcomplex).