Questions regarding basic definitions on CW complexes?

Question

I am reading about CW complexes. My book uses the following definitions.

Definition: A cell complex $X$ is a Hausdorff space which is the union of disjoint subspaces $e_{\alpha} (\alpha \in \mathcal{A})$ called cells satisfying:

a) To each cell we associate an integer $n \geq 0$ called its dimension. If $e_{\alpha}$ has dimension $n$ we often use the notation $e_{\alpha}^n$ for this cell. We write $X^n$ for the union of all cells $e_{\alpha}^k$ with $k \leq n$. $X^n$ is called the $n$-skeleton.

b) If $e_{\alpha}^n$ is an $n$-cell, there is a "characteristic map" $\chi_{\alpha}: (B^n, S^{n-1}) \to (X, X^{n-1}) $ such that $\chi_{\alpha} |_{B^n - S^{n-1}}$ is a homeomorphism from $B^n - S^{n-1}$ onto $e_{\alpha}^n$.

Definition: If $A \subset X$ is a subset of $X$, we define $K(A)$ to be the intersection of all subcomplexes containing $A$. If $A \subset B$, $K(A) \subset K(B)$. Hence if $p \in e, K(p) = K(e) =K(\overline{e})$. Thus $K(A)$ is a subcomplex.

Definition:

C: $X$ is said to be closure finite if for each cell $e_{\alpha}^n$, $K(e_{\alpha}^n)$ is a finite subcomplex.

W: $X$ is said to have the weak topology if for each subset $F \subset X$, $F$ is closed iff $F \cap \overline{e}_{\alpha}^n$ is compact for each cell $e_{\alpha}^n$.

A cell complex satisfying (C) and (W) is called a CW complex.

Questions:

1. Why is $K(e) = K(\overline{e})$? I assume that $\overline{e} = \chi(B^n)$, so I suppose any complex containing $e$ is equipped with this characteristic map and therefore contains $\overline{e}$? Is this reasoning sound?

2. I can't see why exactly it follows from the above that $K(A)$ is a subcomplex. Could anyone provide a few more details on how to see that?

3. I've seen in some other sources that closure finiteness is defined as follows: for every cell $e$ there exist finitely many cells $e_1, \ldots, e_m$, such that $\overline{e} \subset e_1 \cup \ldots \cup e_m$. How to formally show that the conditions are equivalent?

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Edit: As for question $(3)$, I found the answer here: https://math.stackexchange.com/a/4137831/726980

Answer 1

1. If $A$ is a subcomplex of $X$ containing a cell $e$, it also contains $\overline{e}$ by definition. The converse holds obviously. Thus, the set of subcomplexes of $X$ containing $e$ and the set of subcomplexes of $X$ containing $\overline{e}$ are the same and the intersection of these subcomplexes are $K(e)$ and $K(\overline{e})$ respectively.

2. First, you can argue that $K(K(A))=K(A)$ for any subset $A\subseteq X$. Now, if $p\in K(A)$ is arbitrary and $p\in e\subseteq X$ is the cell containing $p$, it follows that $K(e)=K(p)\subseteq K(K(A))=K(A)$, hence $e\subseteq K(e)\subseteq K(A)$. It follows that $K(A)$ is a union of cells. Furthermore, if $e\subseteq K(A)$, then $K(\overline{e})=K(e)\subseteq K(K(A))=K(A)$, hence $\overline{e}\subseteq K(A)$. Thus, $K(A)$ is a subcomplex.

Note, for the record, that it is true in general that the intersection of a collection of subcomplexes is a subcomplex, which you could also argue directly.