Let $X$ be a CW complex. I need to show that $X$ is finitely generated, which means that it's topology is coherent with its compact subspaces.
A space $X$ is said to be compactly generated if:
A subspace $A$ is closed in $X$ if and only if $A ∩ K$ is closed in $K$ for all compact subspaces $K ⊆ X$.
So i'm having trouble getting started here. I suspect that the topology of a CW complex is very connected to the characteristic maps for each open cell. Before we get going here, I have a question: Is an open cell of a CW complex necessarily open in the topology of the CW complex? I want to say it is, because it is homeomorphic to the interior of a disk, but somebody told me it wasn't. Can somebody offer some insight here? Thanks!