Topological spaces similar to but not actual manifolds I'm looking for topological spaces that are similar to but not actual manifolds.
Is there a way to construct topo. spaces that are locally homeomorphic to, say, $S^n$ (rather than $\mathbb R^n$)? By this I mean that every point in my space $X$ has an open neighborhood which is homeomorphic to (in this example) $S^n$.
For example, I can think of gluing circles into a torus where the open sets are the circles that get glued together.
In this case, every point has an open neighborhood (which is $S^1$ by construction), which is not homeomorphic to $\mathbb R$. Note that in this topology, small "patches" are not open sets, only combinations of "cylindrical slices" of the torus. Therefore, there is no open set in my $T^2$ that is homeomorphic to $\mathbb R^n$.
However, this space is not Hausdorff (not even $T_0$), which is a property that I would like to preserve.
Is it possible to construct such Hausdorff spaces?
How similar would they be to typical manifolds?
 A: Every such space is homeomorphic to the product of $S^n$ and a discrete space, i.e. a disjoint union of spheres, in particular, a manifold in the common sense.
The moral: Use noncompact models for your open subsets (if you want to work with Hausdorff spaces).
But, there is an interesting variation on the theme of manifolds where local models are compact (say, spheres of a fixed dimension): Replace open subsets by closed subsets. Thus, you require your space to be a union of (overlapping) topological spheres. To make things more interesting, you require the transition maps to belong to some small subgroup $G$ of the group  of homeomorphisms of the sphere. There is a remarkable class of spaces obtained this way if $G$ is a finite reflection group. The spaces are called buildings. They first appeared in the classical projective geometry and the general theory was developed by Jacques Tits.
A: The relation "locally homeomorphic" is a transitive relation. Note that $\mathbb{R}^n$ is locally homeomorphic to $\mathbb{S}^n$ and vice versa. Therefore, for any topological space $Q$, $Q$ is locally homeomorphic to $\mathbb{R}^n$ iff $Q$ is locally homeomorphic to $\mathbb{S}^n$.
In fact, for any $n$-manifold $M$ with at least one point, $M$ is locally homeomorphic to $\mathbb{R}^n$ and vice versa. Therefore, $Q$ is locally homeomorphic to $M$ iff $Q$ is locally homeomorphic to $\mathbb{R}^n$.
So the project of studying topological spaces which are locally homeomorphic to a manifold $M$ is in fact the project of studing topological spaces which are locally homeomorphic to $\mathbb{R}^n$.
Edit: this answer is no longer applicable since it does not use the OP's definition of "locally homeomorphic".
A: You can certainly define structures that are locally homeomorphic to something different than $\mathbb{R}^n$, for example you may have complex manifolds where the local charts are homeomorphic to $\mathbb{C}^n$, or you can have Banach or Hilbert manifolds where the local charts are homeomorphic to a Banach or a Hilbert space respectively.
However, in general, it makes little sense from a differentiable point of view to require the local homeomorphism to be with respect to a (topological) space that does not have a linear structure, since the linear structure is the one that allows you to have difference quotients and, hence, to have differential calculus (Absolute calculus on manifolds started with this aim with the Italian school of Ricci Curbastro, Bianchi, Levi Civita and so on...).
Take away: yes, in general you can, but it makes little sense if the goal is to look for a notion of derivative outside $\mathbb{R}^n$.
