# How to prove that the two topologies on an embedded submanifold are actually the same one?

Let $$M,N$$ be two smooth manifold and $$F:M\rightarrow N$$ be an injective smooth immersion. We know that with the subspace topology, if $$F:M\rightarrow F(M)$$ is a homeomorphism, then $$F(M)$$is an embedded submanifold of $$N$$.

However, we can also equip $$F(M)$$ with such a topology: $$A\subset F(M) \text{ is open if and only if }F^{-1}(A) \subset M \text{ is open }$$

My professor said that the two topologies are actually the same one when the case is submanifold. But I still confused about why? How can I prove this fact?

This is almost literally the definition of a homeomorphism. To say that a bijection $$f:X\to Y$$ is a homeomorphism means that for any $$A\subseteq Y$$, $$A$$ is open in $$Y$$ in iff $$f^{-1}(A)$$ is open in $$X$$. Applying this to $$F:M\to F(M)$$ says that $$F$$ is a homeomorphism iff the topology you describe is the same as the original topology on $$F(M)$$.