A while ago I asked a <a href="http://mathoverflow.net/questions/18505/what-is-enough-to-conclude-that-something-is-a-cw-complex">question about recoqnizing CW complexes</a> 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). **Question**: Are there some natural topological conditions to put on $X'$ such that it can be proved 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 except the first, and that comes from the previous question): 1) compact Hausdorff (does not work alone). 2) $X'$ is locally contractible. 3) $X'$ has the homotopy type of a CW complex. 4) Combinations of the above. 5) $X'$ 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?