Manifolds and Topological Spaces 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

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.
