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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?

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?

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 except the first):

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?

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?

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).

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?

Added: there are obvious counter examples to 2-5) if 1) is not assumed, so assume this also in 2) through 5). I put it on the list instead of in the assumptions because it may be implied by what ever conditions you might come up with.

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 except the first):

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?