Let $(M,\mathfrak{X})$ be a smooth manifold, where $M$ is a topological manifold and $\mathfrak{X}$ a smooth structure on it.
It is commonly mentioned (in introductory resources on smooth manifolds) that there may exist a (finite or infinite) number of other smooth structures $\mathfrak{X}'$ on $M$ such that $(M,\mathfrak{X}')$ is not diffeomorphic to $(M,\mathfrak{X})$; and that in some special cases, some kind of uniqueness does hold (e.g. for the three-dimenional case, cf. this M.O post).
I am wondering about the analogous question in the category of topological manifolds (i.e., about something analogous to the notion of "exotic structures", but for topological manifolds).
Namely:
Let $(X,\mathfrak{T})$ be a topological manifold (where X is a set and $\mathfrak{T}$ is a topology on $X$ that makes it a topological manifold). In general, does there exist another topology $\mathfrak{T}'$ on $X$ such that $(X,\mathfrak{T}')$ is a topological manifold that is not homeomorphic to $(X,\mathfrak{T})$ ? If so, are there more restrictive situations where we still have uniqueness of the topological-manifold structure ?
Best Answer
If the only thing you're preserving is the set $X$, then you can biject it to the set of points in some other manifold and use that manifold's topology.
For example, suppose you start with $(\mathbb{R},\mathfrak{T})$ where $\mathfrak{T}$ is the standard topology on $\mathbb R$. Then set $f:\mathbb{R}^2\to\mathbb{R}$ to be your favorite bijection (perhaps as in MJD's answer to Examples of bijective map from $\mathbb R^3\to\mathbb R$).
If $\mathfrak{T}'$ is the standard topology on the plane $\mathbb R^2$, then we can turn it into a topology $\mathfrak{T}''$ on $\mathbb R$ by using the forward image under $f$:
$$\mathfrak{T}''=\left\{f[U]\mid U\in\mathfrak{T}'\right\}$$
Now $(\mathbb R,\mathfrak{T}'')$ (the plane encoded so the points are technically real numbers) is not homeomorphic to $(\mathbb R,\mathfrak{T})$ (the real line).
(Instead of the plane, I could have used another 1-d manifold like the circle, but I figured changing dimension would be more striking/make it clear how little "$X$ is the same" controls.)