0
$\begingroup$

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)

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.