Subobjects and quotient objects in category of topological spaces 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. $e_\circ f_1=e\circ f_2 \implies f_1 =f_2$.

and 

$e:A \rightarrow X$ 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.

The first definition actually characterises injective continuous maps, but the second definition does indeed characterise topological embeddings. More generally there is a notion of initial topology and final topology for a family of maps, and an embedding is precisely an injective map that induces the initial topology on its domain, and a quotient is precisely a surjective map that induces the final topology on its codomain. The universal property you have written down is a translation of that into the language of category theory.
