Definition 2.4.6 of the book Topology An Introduction by Stefan Waldmann states the following:
Let $f:(M, \mathcal{M}) \longrightarrow(N, \mathcal{N})$ be a map between topological spaces.
- $f$ is called a homeomorphism if $f$ is bijective, continuous, and if $f^{-1}$ is continuous.
- If there is a homeomorphism $f:(M, \mathcal{M}) \longrightarrow(N, \mathcal{N})$ then the spaces $(M, \mathcal{M})$ and $(N, \mathcal{N})$ are called homeomorphic.
- $f$ is called an embedding if $f$ is injective and if
$$f:(M, \mathcal{M}) \longrightarrow\left(f(M),\left.\mathcal{N}\right|_{f(M)}\right)$$
is a homeomorphism
What confuses me in the above definition is the last point. In the definition of $f$ the input is a topological space $(M,\mathcal{M})$, but in the last point it seems the input is a set $M$. Also, what does the notation $\left.\mathcal{N}\right|_{f(M)}$ mean?
Best Answer
By definition, $\left.\mathcal{N}\right|_{f(M)}$ is the trace of the topology $\cal N$ on $f(M)$, defined by $$ \left.\mathcal{N}\right|_{f(M)} = \{ U \cap f(M) \mid U \in {\cal N}\} $$ Thus $\left(f(M), \left.\mathcal{N}\right|_{f(M)}\right)$ is a well-defined topological space.