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The book "The Topology of CW Complexes" mentions the following result in passing, but does not provide a proof. Let $(X, mathcalC)$ be a CW complex, $B_k subset mathbbR^k$ be the origin-centered open unit-$k$-ball, and $overlineB_k$ be the corresponding closed $k$-ball. Let $f, g : overlineB_k to X$ both be characteristic functions of the same cell $C in mathcalC$. Then there exists a homeomorphism $h : overlineB_k to overlineB_k$ such that $f = g circ h$. How can you prove this statement?

For a regular cell, the statement is of course trivial, so I am specifically interested in the non-regular case.

EDIT

In the light of the counter-example given by Eric Wofsey, I may have misunderstood what is said in the book. The direct quotes are as follows ($dotE_n$ is the boundary of a closed cell $E_n$).

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Page 1:

A closed Euclidean $n$-cell $E^n$ is a homeomorphic image of the Euclidean n-cube $I^n$, the cartesian product of $n$ copies of the closed unit interval $I = t in mathbbR : 0 leq t leq 1$.

Page 6:

Let $X$ be a set. A cell structure on $X$ is a pair $(X, Phi)$,
where $Phi$ is a collection of maps of closed Euclidean cells into $X$
satisfying the following conditions.

1. If $phi in Phi$ and $phi$ has domain $E_n$, then $phi$ is injective on $E_n - dotE_n$.




2. The images $phi(E_n - dotE_n) : phi in Phi$ partition $X$, i.e., they are disjoint and have union $X$.




3. If $phi in Phi$ has domain $E_n$, then $phi(dotE_n) subset bigcup psi(E_k - dotE_k) : psi in Phi text has domain E_k text and k leq n - 1$

Page 7:

We say that two cell structures $(X, Phi)$ and $(X, Phi')$ are strictly equivalent if there is a one-to-one correspondence between $Phi$ and $Phi'$ such that a characteristic function with domain $E_n$ corresponds to a characteristic function with domain $E_n$, and corresponding functions differ only by a reparametrization of their domain. That is, if $phi$ and $phi'$ are corresponding functions of $Phi$ and $Phi'$, respectively, then $phi' = phi circ h$, where $h: (E_n, dotE_n) to (E_n, dotE_n)$ is a homeomorphism of pairs. We leave it to the reader to check that this is an equivalence relation on the collection of cell structures on the set X.

If $(X, Phi)$ is a cell structure, let $mathcalS_Phi$ consist of all pairs $(sigma^n, [phi])$, where $sigma^n = phi(E^n)$ and $[phi]$ is the strict equivalence class of $phi in Phi$. For convenience, we will denote such a pair by $sigma^n$ or $phi(E^n)$ with the class of $phi$ understood. Note that if $(X, Phi)$ and $(X, Phi')$ are strictly equivalent cell structures, then $mathcalS_Phi = mathcalS_Phi'$.

Page 8:

A cell complex on a set $X$ or a cellular decomposition
of a set $X$ is an equivalence class of cell structures $(X, Phi)$ under
the equivalence relation of strict equivalence.

A cell complex on $X$ will be denoted by a pair $(X, mathcalS)$, where $mathcalS = S_Phi$ for some representative cell structure $(X, Phi)$. The set $mathcalS$ is called the set of (closed) cells of $(X, mathcalS)$.

Page 41:

A Hausdorff space $X$ is a CW complex with respect to a family of cells $mathcalS$ provided:

1. the pair $(X,mathcalS)$ is a cell complex such that each cell $sigma in mathcalS$ has a continuous characteristic function;




2. the space $X$ has the weak topology with respect to $mathcalS$;




3. the cell complex $(X,mathcalS)$ is closure finite.

END QUOTES

I'm not sure where my misunderstanding is, but I suspect it has to do with that the strict equivalence is defined between cell-structures and not between characteristic functions.

EDIT 2

Referring to the discussions with Paul Frost below, the confusion arises because the definition in the book is more granular than the usual definition of a CW complex. Therefore, if characteristic functions are not related by a homeomorphism in this definition, then they are not the same CW complex.

In contrast, the usual definition of a CW complex requires only the existence of a characteristic function, which makes it possible to provide non-homeomorphically related characteristic functions for the same CW complex, as provided by the counter-example in the accepted answer.

This is false. If $k>1$, then there exists a map $f:overlineB_ktooverlineB_k$ which restricts to a homeomorphism $B_kto B_k$ and maps $partial B_k$ to itself but which is not injective on $partial B_k$. So, taking $X=overlineB_k$ with a CW-complex structure that has $B_k$ as a $k$-cell, both $f$ and $g=1_overlineB_k$ are possible characteristic functions for the $k$-cell. Since $f$ is not a homeomorphism, there cannot exist any homeomorphism $h:overlineB_ktooverlineB_k$ such that $f=gcirc h$.

For an explicit example of such an $f$ for $k=2$, let $S=[0,1]times[0,1]$ and $T=(x,y)inmathbbR^2:0leq yleq xleq 1$ and define $f_0:Sto T$ by $f_0(x,y)=(x,xy)$. Then $f_0$ is a homeomorphism on the interiors of $S$ and $T$ but collapses one of the sides of $S$ to a single point. Composing with homeomorphisms $S,Tcong overlineB_2$ we get $f:overlineB_2tooverlineB_2$ which is a homeomorphism on the interior but collapses an arc on the boundary to a point.