I am trying to proof the following two statements.
- If $X_1 \subset \dots \subset X_i \subset \dots$ is a infinite sequence of CW-complexes, then $X = \bigcup X_i$ is a CW-complex and each $X_i$ is a subcomplex of X.
edit 0: As Lee Mosher pointed out, (1.) is false in general. But: I am interested in the following case: All $X_i$ are homotopy equivalent to submanifolds $M_i$ of a "bigger" manifold M, such that $M_i \subset M_{i+1}$. What I am really trying to understand is, why M is homotopy equivalent to a CW complex.
Should not be a hard! I tried to prove this by defining the filtration of X via $X^{(n)} = \bigcup X_i^{(n)}$, where $X_i^{(n)}$ is the n-skeleton of $X_i$, equipped $X^{(n)}$ with the weak topology and pick the weak topology induced by the filtration on X. Is this the right approach?
- Why is the tabular neighborhood of a manifold a CW complex? Is there a quick argument? A reference where I can read about it?
(Both questions came up while studying Milnor's Morse Theory proof of 3.5. last part)
edit 1: I guess, I found a way to tackle the seconde question: Essentially, one has to prove is that any open subset of $\mathbb{R}^n$ is a CW-complex. This can be done by subdividing the set into cubes, approximating it with smaller and smaller cubes near the boundray. Since the tabular neighborhood is an open set, this proves 2.
This is not true, not without some extra conditions.
For one counterexample, take $X_i \subset \mathbb{R}^2$ to be the union of the circles of radius $1/j$ centered at the points $(1/j,0)$ for $1 \le j \le i$. The union is the famous Hawaiian Earring, which is not a CW complex.