Question about Hatcher's definition of CW-complex In page 519 of Hatcher's book, we can find the following definition of CW complex:

[...] Let us first recall from Chapter 0 that a CW complex is a space $X$ constructed in the following way:

*

*Start with a discrete set $X^0$, the $0$-cells of $X$.

*Inductively, form the $n$-skeleton $X^n$ from $X^{n-1}$ by attaching $n$-cells $e_\alpha^n$ via maps $\varphi_\alpha:S^{n-1}\to X^{n-1}$. This means that $X^n$ is the quotient space of $X^{n-1}\bigsqcup_{\alpha}D_\alpha^n$ under the identifications $x\sim \varphi_\alpha(x)$ for $x\in\partial D_\alpha^n$. The cell $e_\alpha^n$ is the homeomorphic image of $D_\alpha^n-\partial D_\alpha^n$.

*$X=\bigcup_nX^n$ with the weak topology: A set $A\subset X$ is open iff $A\cap X^n$ is open in $X^n$ for each $n$.


Then, Hatcher remarks:

Note that condition (3) is superfluous when $X$ is finite-dimensional, so that $X = X^n$ for some $n$. For if $A$ is open in $X = X^n$, the definition of the quotient topology on $X^n$ implies that $A\cap X^{n-1}$ is open in $X^{n-1}$, and then by the same reasoning $A\cap X^{n-2}$ is open in $X^{n-2}$, and similarly for all the skeleta $X^{n-i}$.

My question is: Is $X=\bigcup_nX^n$  the classical union of sets or it should be understood as the colim of the sequence $X_1\to X_2\to X_3\to\ldots$?
I can't understand how it can be deduced that if the union is finite then $X=X^n$.
 A: The technically "clean" way is to take the colimit of the sequence of spaces.
In fact, the process of attaching cells is a special of the adjunction space construction: Given spaces $X,Y$ and a map $f : A \to Y$ defined on a subspaces $A \subset X$, we define the adjunction space $X \cup_f Y$ as the quotient space of the disjoint union $X + Y$ obtained by identifying each $a \in A$ with its image $f(a) \in Y$. This yields a canonical embedding $i : Y \to X \cup_f Y$.
But is $Y$ a genuine subspace of $X \cup_f Y$? This is a somewhat philosophical question and I think the answer is irrelevant. The purpose of the adjunction space is best explained by its universal property as a pushout; see here.
Anyway, the answer depends on the specific construction of $X \cup_f Y$. On the level of sets the standard construction of $X \cup_f Y$  produces a set of equivalence classes in $X + Y$, i.e. a subset of the power set of $X + Y$ which never contains $Y$ as a genuine subset (whatever the concrete definition of $X + Y$ may be). But with some work one can also give an alternative construction: Find a bijection $\beta : X \to X'$ to a set $X'$ such that $X' \cap Y = \emptyset$, then take $X \cup_f Y = (X' \setminus \beta(A)) \cup Y$ and give it the the appropriate topology. Then $Y \subset X \cup_f Y$.
Let us come back to CW complexes. $X^n$ is obtained as an adjunction space  from $X^{n-1}$ by attaching a collection of $n$-cells. We certainly get a canonical embedding $i_n : X^{n-1} \to X^n$; but whether $X^{n-1}$ is a genuine subspace of $X^n$ depends on the chosen construction. Yes, usually one writes $ X^{n-1} \subset X^n$, but some caution should be taken.
I think the direct limit approach makes it conceptually clearer why we topologize infinite-dimensional CW-complexes by giving them the weak topology; again there is a universal property.
Note that if we have an ascending sequence of spaces $X^0 \subset X^1 \subset X^2 \subset \ldots $, then the union $\bigcup_n X^n$ can be given the weak topology as in 3.  and the resulting space has the universal property of the direct limit.
