A while ago I asked a question about recoqnizing CW complexes and got an extremely nice and concrete answer. However, I am still interested in a more general treatment of this and therefore pose the following closely related question:
Assume that $X$ is an $n−1$ dimensional finite CW complex. and assume that $X'$ is given as a a set by the disjoint union of $X$ and a single open cell $e$ of dimension $n$. I.e. $e$ is an open subspace of $X'$ homeomorphic to the open $n$ disc (and of course $X$ is homeomorphic to the complement). Also assume that $X'$ is compact Hausdorff.
Question: Are there some natural topological conditions to further put on $X'$ such that it follows that $X'$ is in fact a CW-complex with $X$ as a sub-CW complex and $e$ a single new cell?
Remark: The fact that $X$ is such is equivalent to whether or not one can modify the homeomorphism of $e$ to the open unit disc such that the inverse extends to the closed unit disc.
Ideas on answers (but I have no proofs or counter examples in any of the cases):
1) $X'$ is locally contractible.
2) $X'$ has the homotopy type of a CW complex.
3) The Combination of 1) and 2).
4) $X'$ is homeomorphic to a CW complex
The last is weird, but it is not clear to me that even this is enough!
Any ideas, counter examples, or references?
