Ytukyg

Let $X$ be a CW-complex. I want to prove each of its path components contains a zero cell. For that, I want to prove that each of its components is again a CW-complex.

Please give me some hints.

Let C be your pathcomponent, and take any cell $e_nto C$, which is, say, of dimension $n$. Then the boundary of $e_n$ maps to a cell of dimension ___?

A subcomplex of a CW-complex $X$ is a subspace $A subset X$ which is a union of closed cells of $X$. Recall that a closed $n$-cell $e^n_alpha$ is the image of an attaching map $phi^n_alpha : D^n to X^{n-1}$ defined on the closed unit ball in $mathbb{R}^n$. In particular, each $e^n_alpha$ is pathwise connected. Note that if $e^n_alpha subset A$, then also all $e^m_beta$ such that $mathring{e}^m_beta cap e^n_alpha ne emptyset$ must be contained in $A$. Here $mathring{e}^m_beta$ denotes the open cell associated to $e^m_beta$, i.e. the set $phi^m_beta(mathring{D}^m) subset e^m_beta$. $mathring{D}^m$ is the open unit ball in $mathbb{R}^n$.

Each path component $P$ of $X$ is a union of closed cells: If $e^n_alpha cap P ne emptyset$, then $e^n_alpha cup P$ is pathwise connected so that $e^n_alpha subset e^n_alpha cup P subset P$.