Definition: A subset $N \subset M$ has the submanifold property if for all $p$ in $N$, there exists a coordinate neighborhood $(U, \varphi)$ such that:
- $\varphi(p)=0$
- $\varphi(U)= (-\varepsilon, \varepsilon)^n$
- $\varphi(U \cap N) = \{x \in (-\varepsilon, \varepsilon)^n \; | \; x_{m+1}= \ldots = x_n=0 \}$
A regular submanifold is a subset $N$ with the submanifold property and with the topology induced by $M$.
Lemma: Let $M, \; M'$ be two smooth manifolds and $f: M' \longrightarrow M$ be a smooth map. If $N \subset M$ is regular submanifold, then $f': M' \longrightarrow N$ is also smooth.
Proof: $f: M' \longrightarrow M$ and $f':M' \longrightarrow N$ only differ by the projection, which is smooth.
Why is the regularity essential in this lemma?
Suppose you put the discrete topology on $N$, i.e. all subsets of $N$ are open. Then the map $f': M \to N$ is not even continuous, let alone smooth!
To elaborate: The submanifold property (as defined above) only says that $N$ sits inside $M$ as a nice subset. Since the topology on $N$ is not specified, we cannot make statements about continuity or smoothness, which depend on the topology.