I was trying to understand the definition of CW complex which is given in Hatcher Appendix, that is the one given in this previous question: Understanding Hatcher's definition of CW-complex
In doing that, I was wondering if a n-1 skeleton could be topologically embedded in a n skeleton. More precisely, let $\pi:X^{n−1} \cup \cup_{\alpha} D_{\alpha}^n\rightarrow X^{n}$ be the natural projection and $i:X^{n-1}\rightarrow X^{n−1} \cup \cup_{\alpha} D_{\alpha}^n$ be the inclusion, I was wondering if $\pi\circ i:X^{n-1}\rightarrow X^{n}$ is an homeomorphism on the image(with the subspace topology). I would say yes: it is a right supposition?
If I wasn't mistaken, I showed that is injective and continuous, but I failed in showing for example tha it is open as map on the Image
Thanks for the help