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?
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Best Answer
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 $F(M)$.