Manifolds and Topological Spaces

Question

from my understanding of manifolds they are structures defined on topological spaces. So if M is a manifold defined on a topological space $(X,\tau)$ and $X\subseteq\mathbb R^3$, does this mean $M$ is a $3$-manifold? If so does this generalize to higher dimensions such as if $X\subseteq\mathbb R^n$ would that mean $M$ is an $n$- manifold? - thanks

Answer 1

A manifold is a topological space that locally looks like Euclidean space. The topological space does not need to be a subspace of some $\mathbb{R}^n$, but even if it is, its dimension need not be $n$. For example, the sphere is $2$-dimensional but naturally sits in $\mathbb{R}^3$. The key point here is that every point on the sphere has a neighborhood that looks like an open subset of the plane $\mathbb{R}^2$. Put less rigorously, there are only two free directions of movement on the sphere.

Answer 2

It is usually said that the notion of manifolds was introduced by Riemann in 1854, but it wasn't until Whitney’s work in 1936 that people know what abstract manifolds are, other than being submanifolds of Euclidean space. A plane in R^3 is a very trivial counterexample of your question.