Are the following definitions are equivalent?
An embedding of topological spaces is just a monic in Top, i.e. it is a continuous $e: A \rightarrow X$ s.t. for any space $Z$ the function $f \mapsto e \circ f:$ Top $\left(Z,A\right)\rightarrow$ Top $\left(Z,X\right)$ is injective i.e. $e_\circ f_1=e\circ f_2 \implies f_1 =f_2$.
$e:A \rightarrow X$ is an embedding iff
- $U\left(e\right):U\left(A\right)\rightarrow U\left(X\right)$ is injective in Set, where $U$ is the underling set functor.
- for any $Z \in$ Top a set map $U\left(f\right):U\left(Z\right) \rightarrow U\left(A\right)$, $f$ is continues iff $e \circ f : Z \rightarrow X$ is continuous.
Is the analogue definitions for epic and quotient space are also equivalent? Many thanks.