I find it hard to understand the definition of $CW$ subcomplex(and even $CW$ complex) when it comes to examples. For instance, is it possible to view $S^k\times\{0,1\}$ as a $CW$ subcomplex of $S^k\times I$ by defining a proper $CW$ structure on $S^k\times I$? Is it legal to construct $I$ as a CW complex by "attaching a 1-cell to simultaneously two discrete points $0$ and $1$"?
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$I=[0,1]$ can be given a CW complex structure as follows:
- 0-skeleton: $X^0 = \{0, 1\}$. By definition 0-skeleton has to be discrete. So we are fine. I will denote members of $X^0$ by $0_{X^0}$ and $1_{X^0}$ respectively.
- 1-skeleton: $X^1 = [0, 1]$ constructed by attaching $[0,1]$ (note that $[0,1]$ is a 1-dimensional closed ball, i.e. a cell) to $X^0$ with trivial identification $0_{X^0}\sim 0$ and $1_{X^0}\sim 1$. There's only one 1-dimensional open cell: $(0, 1)$.
Also note that there are multiple possible CW structures on $[0,1]$ (actually there are infinitely many possibilities) but this one is particularly easy being the smallest one (in terms of number of cells).
From that point of view $A=\{0, 1\}$ is an obvious subcomplex. By definition it is simply $X^0$ and a closure of each cell (being 0 dimensional cell) is in $A$.
So now you just need to recall how CW structure is generated on the product of CW complexes. See this: Cartesian product of two CW-complexes and how these two definitions play together.
