Embedding onto adjunction space

Question

Let X and Y be topological spaces. Let A $\subset$ X be a closed set and $f : A \to Y$ a continuous map. Prove that the canonical map $Y \to Y \cup _fX$ is an embedding onto a closed subspace.

$Y \cup _fX$ denotes the adjunction space.

What do I exactly have to prove as I don't quite understand the concept of an embedding?

Answer 1

Prove that the canonical map $Y \to Y \cup _fX$ is an embedding onto a closed subspace.

Denote this map by $g$. You have to prove the following:

• the map $g$ is injective;

• the map $g$ is continuous;

• the inverse map $g^{-1}:f(Y)\to Y$ is continuous;

• the image $g(Y)$ is closed in $Y \cup _fX$.