Can we glue CW complexes to themselves?

Question

My understanding: CW complexes are constructed by inductively attaching $n+1$ cells to an $n$-skeleton where a 0-skeleton consists of a discrete set of 0-cells (points).

What I fail to understand: real projective space $\mathbb{R}\mathrm{P}^n$ is constructed by using pairs of $k$-cells and joining the two resulting "hemispheres" together and then gluing these two pieces together properly. Doesn't this construction break the rules of CW complexes? I have no problem attaching $n+1$ cells to lower order cells. Are we allowed to glue CW complexes to themselves though?

Edit: hopefully this illustrates what I mean by "gluing a complex to itself". We start out with two 0-cells:

$X_0=\{-1,1\}$

We attach two 1-cells, one going from -1 to 1 and the other going from 1 to -1.

$X_1=\{(-1,1),(1,-1)\}$

(Please forgive my notation, I know it's awful so bear with me) We now have a topological space homeomorphic to $S^1$. The final step is to glue either of the 1-cells to the other and the result is $\mathbb{R}\mathrm{P}^1$. This step is what my question is about.

Edit 2: to make the "gluing" step more precise, I mean that we identify antipodal points with one another and take the quotient space as our new real projection space. This construction generalises to higher dimensions.

Answer 1

What I failed to understand is that:

• $n$-cells can be attached in non-obvious ways.
• the transition from $\mathbb{R}\mathrm{P}^{1}$ to $\mathbb{R}\mathrm{P}^{2}$ (i.e. $S^1$ $\to$ Boy's surface) requires attaching the 2-cell in just such a way.

So basically, "self-gluing" is not permitted in the construction of CW complexes per se, but it is possible to construct CW complexes that can be interpreted as "self-glued" (as is the case with real projective space).