In Hatcher's Algebraic Topology, he defines a CW complex as follows:
He proves in Proposition A.1. that any space constructed this was will satisfy the closure finiteness property, and then gives an example of a cell complex which doesn't:
My question is, how is this a counterexample? It seems to me that this can be constructed in the way he presents in his CW complex definition. Can someone explain to me what's going on?
In part (2) of the definition, "map" means continuous map, so the attaching maps $\varphi_\alpha: S^{n-1}\to X^{n-1}$ are required to be continuous. This is not true in the counterexample: the $0$-skeleton is $S^1$ with the discrete topology (by part (1) of the definition), and so the identity map $S^1\to X^0=S^1$ which would attach the $2$-cell is not continuous (the domain has the usual topology of $S^1$ but the codomain has the discrete topology).