Definition of subcomplex of a CW-complex I am studying Algebraic Topology from Hatcher, they define a subcomplex of a cell complex X as a closed subspace A of X such that A is a union of cells of X.
I understend this definition, but for me it's counter-intuitive that a subcomplex A of X depends on the cellular structure of X.
For example if we take X=$S^2$ the sphere, it can be seen as a cw-complex with two different cellular structures:

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*It's a union of one $0$-cell and one 2-cell

*It's a union of one $0$-cell, one $1$-cell and two $2$-cell

If then I take A the $1$-cell of the second structure, it is a cw-subcomplex of X with the second structure but it is not a cw-subcomplex of X with the first structure.
Why don't we have a definition of a subcomplex which is indipendent from the cellular structure of X?
 A: Hatcher is a bit sloppy when he introduces the concept of a CW-complex. He says that a CW-complex is a topological space constructed inductively by a certain procedure.
This suggests that a CW-complex is nothing else than a topological space which admits a certain decomposition into cells of various dimensions, but without reference to a specific cell decomposition.
Note, however, that at some places Hatcher uses the phrase CW-structure to refer to a specific cell decomposition.
If a CW-complex is interpreted as a topological space of a certain type, then your doubts are absolutely justified. In general there are many different CW-structures on the same space $X$ and the notion of a cell of $X$ does not make any sense.
This shows that it is not expedient to understand a CW-complex just as a topological space of a certain type. It is much more adequate to understand a CW-complex as a pair $(X,\mathfrak C)$ consisting of a space $X$ and a CW-structure $\mathfrak C$ on $X$. This is the standard concept of a CW-complex.
Doing so nicely explains the definition of a subcomplex of a CW-complex.
Actually also Hatcher uses the above concept of a CW-complex.

*

*In the section "Cellular Homology" he works with the skeleta $X^n$ of a CW-complex $X$ (see Lemma 2.34) and with it cells (see for example the Cellular Boundary Formula). Without reference to a CW-structure on $X$ this would not make any sense.


*He introduces the concept of cellular maps between CW-complexes and proves the cellular approximation theorem (Theorem 4.8). Again CW-structures are essential.
