What is a Topological space isomorphic to euclidian space?

Question

Wikipedia says:

a topological manifold is a topological space locally homeomorphic to a Euclidean space.

I understand that the fact that a topological manifold is homeomorphic to euclidian space does not imply that the manifold inherits all properties of euclidian space. For example, it does not mean that the manifold locally has the metric that euclidian space has.

I have two questions relating to this:

1. What is the difference between a general "manifold" and a "topological manifold"?

2. If we would define (informally) a programmer2134-manifold as "a topological space locally isomorphic to Euclidian space." Would that change? What would such a manifold generally be called? And what properties would it inherit from Euclidian space? Would it inherit the Euclidian metric?

Answer 1

Here is an answer to the question that you asked in your comment about whether there exists a more general notion than a topological manifold.

In some specialized situations people study "$\square$-manifolds" where $\square$ is replaced by a certain very special kind of topological space, and where some overlap condition is sometimes required, similar to the condition on a smooth manifold that its overlap maps be smooth.

Here's some examples I am aware of:

• "Banach manifolds" which are locally homeomorphic to a Banach space (with an additional overlap condition).
• "Hilbert cube manifolds", which are locally homeomorphic to the Hilbert cube.
• "Menger manifolds", which are locally homeomorphic to the Menger curve aka the Menger sponge.

The mathematical theories of these different examples are very different from each other. Perhaps their important common feature is that the model space ($\mathbb{R}^n$, Banach space, Hilbert cube, Menger sponge) is highly homogeneous, for example its homeomorphism group acts transitively.

Answer 2

The term 'manifold' here is usually short-hand for 'smooth manifold' so specifically the relationship lies with the fact that a smooth manifold is a topological manifold whose transition maps are all smooth.

That is to say, if a topological space $X$ is such that it has an open cover $\{U_i\}$ with homeomorphisms $x_i:U_i \to \mathbb R^n$, and such that on each intersection $U_i \cap U_j$, the maps $x_j \circ x_i^{-1}:\mathbb R^n \to \mathbb R^n$ are all smooth maps.

The above paragraph shows that this condition is a lot stronger a condition then simply asking for $X$ to be Euclidean. So a topological manifold is just a Space $X$ that is locally Euclidean, possibly Hausdorff.