I am reading Algebraic Topology from Hatcher.
There in page 7, A subcomplex of a cell complex $X$ is defined as a closed subspace $A$ of $X$ such that $A$ is a union of cells of $X.$
I don't get this definition. What is cell of $X?$ Is it $n-$ skeleton?
Please help.Thank you.
Best Answer
Quotation from p.5:
Start with a discrete set $X^0$ , whose points are regarded as $0$-cells.
Inductively, form the $n$-skeleton $X^n$ from $X^{n−1}$ by attaching $\color{red}{\text{$n$-cells}}$ $e^n_\alpha$ via maps $\varphi_\alpha : S^{n−1} \to X^{n−1}$. This means that $X^n$ is the quotient space of the disjoint union $X^{n−1} \bigsqcup_\alpha D^n_\alpha$ of $X^{n−1}$ with a collection of $n$-disks $D^n_\alpha$ under the identifications $x \sim \varphi_\alpha(x)$ for $x \in \partial D^n$. Thus as a set, $X^n = X^{n−1} \bigsqcup_\alpha e^n_\alpha$ where each $e^n_\alpha$ is an open $n$-disk.
One can either stop this inductive process at a finite stage, setting $X = X^n$ for some $n < \infty$, or one can continue indefinitely, setting $X = \bigcup_n X^n$. In the latter case $X$ is given the weak topology: A set $A \subset X$ is open (or closed) iff $A \cap X^n$ is open (or closed) in $X^n$ for each $n$.
A space $X$ constructed in this way is called a cell complex or CW complex.
Thus you see that $X$ is the disjoint union of cells (= homeomorphic copies of open disks) $e^n_\alpha$ of various dimension.