# Possible Generalization of a Manifold

A manifold $M$ is a second-countable, Hausdorff, locally Euclidean topological space. Obviously, there are advantages to requiring $M$ to be locally Euclidean, i.e. in some cases this allows $M$ to be endowed with a smooth structure.

The "locally Euclidean" axiom, outside of the setting it creates for the establishment of a possible calculus on $M$, is just a statement about the relationship $M$ has to a very particular metric space, namely $\mathbb{R}^n$. If one has no interest in doing calculus on $M$, I see no reason to restrict this "relationship" to $\mathbb{R}^n$. This motivates the following definition:

Let $A$ be a second-countable metric space. An $A$-manifold is a topological space $M$ such that $M$ is second-countable, Hausdorff and:

For each $p \in M$ there is an open set $U \subseteq M$ with $p \in U$, an open set $U^* \subseteq A$ and a homeomorphism $\phi: U \rightarrow U^*$.

Locally Euclidean manifolds then become the special case $A= \mathbb{R}^n$ for some $n$. One could of course generalize further, i.e. require $A$ only to be a topological space, but at that point the working mathematician is really only doing general topology.

To anyone's knowledge, have $A$-manifolds, or similar structures, been explored in detail? That is, is there any literature exploring the properties of these or similar structures?

NOTE: Also, if anyone has an objection to my statement: "If one has no interest in doing calculus on $M$, I see no reason to restrict this "relationship" to $\mathbb{R}^n$," please mention it.

• Maybe you could take a look on something which is called orbifold. I'm not an expert in this area but as far as I know these are some sort of $A$ manifolds where $A=\mathbb{R}^{n}/G$ for some group $G$ – truebaran May 15 '16 at 22:30
• Certainly many people are interested in the topology of manifolds in their own right, without a desire to do calculus; this is hardly a 'finished' theory. In any case, topological manifolds with boundary are what you get when you let $A$ be the closed upper half-space in $\Bbb R^n$. @truebaran: Orbifolds are defined with charts and whatnot, and are not really ever considered solely as topological spaces. As a simple case to see why not, consider the group of diagonal matrices $V_4 \subset SO(3)$; $\Bbb R^3/V_4$ is still homeomorphic to $\Bbb R^3$, but is quite different as an orbifold. – user98602 May 15 '16 at 22:39
• Orbifolds are almost the same thing as "What you get when you quotient a smooth manifold by the action of a finite group" (almost, because there are orbifolds that are not actually the quotient of any manifold). This does not really work as motivation in the topological world, when you could get wildly badly behaved quotients, like the Alexander horned ball. – user98602 May 15 '16 at 22:40
• There are a few other spaces $A$ where these concepts have been studied and have important applications. For instance with $A=$the Hilbert cube, one gets Hilbert cube manifolds, which are important in simple homotopy theory. With $A=$a Banach space one obtains Banach manifolds, which are important in topology and analysis. – Lee Mosher May 16 '16 at 2:56
• Ok so what I'm gathering here is that one usually doesn't attack these structures in full generality, typically these are considered on a case-by-case basis. – M10687 May 16 '16 at 10:17