REMARK: I had already posted these questions, about one hour ago, but one of the questions was not what I meant.
I am in the beginning of my studies in Algebraic Topology and am studying CW complexes and identification spaces. I have two questions and I don't know how to approach any of them:
Give an example of an identification space which does not admit a CW structure.
Consider the 2-sphere $S^2$ and let $X\subset S^2$ be a finite subset. Let $S^2/X$ be the quotient space given by the relation $xRy\Longleftrightarrow x,y\in X$. Give a CW complex structure on $S^2/X$.
For the first question, which irregularity should I give to my space so that it does not admit the CW structure? For the second question, should I try to find a space homeomorphic to $S^2/X$?
I would appreciate any hint on the questions! Thank you in advance.
Question 1 has been answered in principle and 2. is quite a nice question!
One way of doing it is to realize $S^2$ as the union of two hemispheres $E^2_+, E^2_-$ with intersection $S^1$. Then give $S^1$ the structure of a polygon with a set $X$ of $n$ vertices. Now each $E^2_+. E^2_-$ can be given a cell structure with $S^1$ as the $1$-skeleton, and one $2$-cell. So we get a cell structure for $S^2$, with $n$ $0$-cells, $n$ $1$-cells, and two $2$-cells.
Now you should be able to see what $S^2/X$ looks like, and what is its cell structure.
More exposition on identification and adjunction spaces, and finite cell complexes, is given in my book Topology and Groupoids.