Induced Topology via Injective Map

Question

Let $M$ be a topological manifold and $X$ a set. Assume that there is an injective map $f: X \to M$. Is it always possible to define a topology on $X$ such that $f: X \hookrightarrow M$ is a topological embedding? Can we do a similar thing for smooth manifolds i.e. an induced smooth structure (and topology!) on $X$ such that $f: X\hookrightarrow M$ is a smooth embedding?

Answer 1

In the topological context, we can. Just define on $X$ the initial topology $\tau$ with respect to $f$: if $A\subset X$, then $A\in\tau$ if and only if $A=B\cap X$, for some open subset $B$ of $M$.

In the case of smooth manifolds, in general we can't. Take $M=\mathbb R$, $X=\mathbb Q$ and $f$ the inclusion of $\mathbb Q$ in $\mathbb R$.