Show that $X\setminus e_0$ is a subcomplex, and X is homeomorphic to an adjunction space obtained from $X\setminus e_0$ by attaching a single n-cell.

Question

Suppose X is an n-dimensional CW complex with n>=1, and $e_0$ is any n-cell of X. Show that $X\setminus e_0$ is a subcomplex, and X is homeomorphic to an adjunction space obtained from $X\setminus e_0$ by attaching a single n-cell.

Can someone tell me how to prove this question? I don't know how to prove this question

Answer 1

You have to check the definition of a CW-complex to prove directly that $X-e_0$ is one. It's not very difficult, the "geometric" remark consists to say that $e_0$ isn't required to attach any other cell (since it has maximal dimension), so you can put it away without disturbing the CW-complex too much.

The homeomorphism part should be straightforward.