For now, I'll just ask about $1$ dimensional CW complexes. As I understand the definition, its something like this:
- $X^0$ is a set of points ($0$ dimensional objects) the $0$ dimensional skeleton
- A set $A^1$ and a corresponding set of new open unit intervals $\{e^1_\alpha\}_{\alpha \in A^1}$
- A continuous mapping, $\varphi^1:(\coprod_{\alpha \in A^1}S^{0}_\alpha) \rightarrow X^0$ from the boundaries of the new open intervals to the previous skeleton. This induces an equivalence relation where $x \sim y$ where $\varphi^1(x)=y$
- The next skeleton is $X^1=\frac{X^0 \cup (\coprod_{\alpha \in A^1}D^{1}_\alpha)}{\sim}$, the quotient space of the previous interval and the new closed intervals.
Q1. Did I get this right?
Assuming I did, my other questions are:
Q2. Are the boundaries of the closed intervals included in $X^1$? It seems like they should be from the definition of $X^1$ but I don't understand why definitions of CW complexes always mention the open intervals as the "important" thing.
Q3. What are the open sets in new intervals? Is it just the subset topology? Does it include the endpoints? E.g. is $[0,0.5)$ an open set? I guess this depends on the previous question.
Q4. How do I sort out the open sets of $X^1$? For example, let's say I build a circle by taking a single point and a single interval and mapping the endpoints of the interval to the one point. An interval containing that point should be open, but how do I get that from $\varphi^1$?
I know this is kind of four questions, but they are really all focused on the same confusion I have. No need to answer them all if you can help get me past my confusion.