CW complex definition ambiguity

Question

I've seen two restrictions for a CW complex, and I am not sure which is right or if they are equivalent.

A space $X$ is a CW complex if it has a partition into open sets $e^n_i$ such that:

• $X^n := \bigcup_{k \leq n} \bigcup_{I \in I_k} e^n_i$. There are maps $\phi_i^n:D^n \to X^n $ such that:

  • $\phi_i^n$ restricts to a homeomorphism on the interior of the disk to the cell $e^n_i$

  • $\phi_i^n(\partial D^n)$ is contained in finitely many $e^k_j$, with $k<n$.

• $X$ has the weak topology on $X^n$.

My question is the following:

I've also seen condition that the closure $\overline{e^n_i}$ be contained in finitely many $e^k_j$, with $k<n$. (The closure of a cell is contained in finitely many cells of lower dimension.) Is this equivalent to the definition given above?

As a somewhat different question: Could someone show me exactly how this definition and the inductive definition (a la Hatcher) are the same? I intuitively understand each as saying the space is made of a series of cells where the boundaries of higher dimensional cells are attached to lower dimensional cells. But, I can't see a specific homeomorphism between the two definitions)

Answer 1

First, your definition is incorrect in several ways:

• The space $X$ must be Hausdorff.
• The sets $e_i^n$ should not be required to be open.
• It's not clear to me what "$X$ has the weak topology on $X^n$" is supposed to mean, but I don't see any interpretation that would be correct. The correct requirement is that $X$ has the weak topology with respect to all of the maps $\phi_i^n$ (that is, a subset $U\subseteq X$ is open iff $(\phi_i^n)^{-1}(U)$ is open for all $i$ and $n$). This is equivalent to $X$ having the weak topology with respect to the subspaces $X^n$ if there are only finitely many $e^n_i$ for each $n$, but not in general.

As for your first question, no, that is incorrect: $\overline{e^n_i}$ cannot possibly be contained in cells of lower dimension, since $e^n_i$ itself is disjoint from the other cells. The correct statement is that $\overline{e^n_i}\setminus e^n_i$ is contained in finitely many $e^k_j$ with $k<n$. This is equivalent to the requirement on $\phi_i^n(\partial D^n)$ since in fact $\phi_i^n(\partial D^n)$ and $\overline{e^n_i}\setminus e^n_i$ are the same set.

As for your second question, you can find a proof in the appendix of Hatcher. Specifically, Proposition A.2 proves that Hatcher's inductive definition is equivalent to your (corrected) definition assuming $X$ is Hausdorff, and Proposition A.3 proves that a CW-complex by Hatcher's definition is always Hausdorff.