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A topological manifold $M$, loosely speaking, is a topological space that is locally homeomorphic to Euclidean space. Here the more formal one:

A topological space $X$ is called locally Euclidean if there is a non-negative integer $n$ such that every point in $X$ has a neighborhood which is homeomorphic to the Euclidean space $E^n$ (or, equivalently, to the real $n$-space $R^n$, or to some connected open subset of either of the two).

First of all, is it correct that this definition assumes the so called "standard topology" on $R^n$ (i.e. whose basis is the set of all open balls at all points)?

More importantly:

Of course a homeomorphism is a topological isomorphism. This means that a Topological manifold only locally inherits the topological structures on $R^n$, not its other structures, such as differentiability, its metric, etc.

while I think I understand what this literally means, I do not fully understand what it does, and what it does not imply about $M$. i.e. what properties of $R^n$ are and are not "contained" in the topological structure that $M$ inherits locally from $R^n$.

  • For instance: Clearly, $M$ does not inherit the Euclidean metric. However, even if we ignore the Euclidean metric, there is an "order" in $R^n$. For example, in a manifold $R^1$, a point $4$ is "between" point $3$ and point $5$. This means that if we start at point $3$, and move "to the right" (or whatever one wants to call it), we will move first through point $4$, then through point $5$. Is this "sequential" property preserved in $M$? and how does the topological structure contain this information? The topology on $R^n$ does not refer to such an order, yet it seems essential to the concept of Euclidian space.

  • Another instance: Even if we ignore the Euclidean metric, there is clearly a "directionality" or concept of "moving away from or towards" a point, in $R^n$. For example, if we start at point $(1,1)$ in $R^2$, and move towards point $(3,4)$, then clearly we are "moving away from" $(0,0)$ we cannot quantify this moving away, without invoking the Euclidean metric. However, is there some more abstracted notion of "directionality" that is retained in the topological structure of $R^n$ and transferred to $M$? If not, then there may be many topological manifolds that look very different from the ones usually pictured as examples.

There may be more such properties I didn't think of.

Something that might help: Is there a simple example of a mathematical structure that is globally homeomorphic to $R^n$, but has none of the other essential properties of $R^n$? That might help clarify which properties are and are not preserved in $M$.

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    In-betweenness (what I assume you mean by "order") is definitely not preserved. ($\Bbb R^1$ has an order. What kind of order are you giving to $\Bbb R^2$?) Neither is directionality, though I think _Riemannian manifolds_ do have a sense of directionality. (Consider the surface of a cube. It's a topological manifold. If directionality were a thing, what would happen if a point were to move towards a vertex? What would "continuing in the same direction" mean when it hits it?) – 2017-02-05

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  1. In addition to the "locally euclidean" property that you mentioned one usually requires a topological manifold to be Hausdorff and 2nd countable. Under these two assumptions, every manifold is homeomorphic to a subset of $R^N$ (one can take $N=2n+1$ where $n$ is the dimension of the manifold).

  2. Yes, one considers $R^n$ equipped with the standard topology.

  3. Neither "betweenness" (whatever it means for $n\ge 2$: I do not think it has any meaning), nor "directionality" (whatever it means) have any meaning for a topological manifold. Take a simple example, like the $n$-dimensional sphere and try to make sense of these two words (I would not even call them notions). Even on the circle, you cannot make sense of the statement that one point is between two other points. (You need orientation for that.)

  4. To check your intuition, just think of the graph of a continuous function $R^2\to R^2$: It is a topological submanifold of $R^4$. But do not assume any degree of smoothness, think of a fractal surface.

  5. Nevertheless, topological manifolds do inherit some properties of $R^n$, such as local connectivity/contractibility, the fact that a bijective continuous map between manifolds is a homeomorphism, etc.

  6. My suggestion is to take Munkres' book "Topology" and read it through the chapter on dimension theory. If you make it there, you will get a much better idea what topological structures/notions are meaningful and which are not.

  7. If you really are a programmer, my suggestion is to think not of topological manifolds but of piecewise-linear manifolds and smooth manifolds.