I am using Arkowitz's book "Introduction to Homotopy Theory", and I have a quite simple question. Arkowitz defines a CW complex in the following manner:
Then he states the following Lemma:
Item (2) is proved as follows
My question is concerning the proof on the second line when he starts the "Conversely". He is taking a subset $F\subset X$ which has the property of being closed in every cell of $X$, thus it is closed in $X$ by definition of CW complex, and he wants to show that $F\cap X^n$ is closed in $X^n$ for each n-skeleton of $X$. But my question is, isn't this obvious from basic topology? If you have a topological space (in this case $X$), a subspace ($X^n$), and a closed subset ($F\subset X$), then $F\cap X^n$ is closed in the subspace $X^n$ by definition.
Am I missing something? I think I probably am.
He says in the definition that $X^n$ is a subspace of $X$, but I think it is confusing. It raises the question if the notion of CW complex is well defined. This is because topology $\tau$ of $X$ is defined as a weak topology with respect to cells and a priori there is no reason why $X^n$ carries subspace topology with respect to $\tau$. So you may consider his proof as showing precisely that this is the case.