Embedding onto adjunction space 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?


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 $Y \cup _fX$.
