Meaning of a discrete topological sub-space? Given a topological space $X$ and a set $U\subseteq X$, what is the meaning of $U$ being a discrete sub-space of $X$?
I do know what a discrete space is, so as far as I understand it, the meaning is that each $A\subseteq U$ is open in the relative topology in $U$? And if I understand it correctly, given $U$ is open in $X$, will this apply that $A$ is open in $X$ as well?
Please fix me if I'm wrong, just wanted to make sure I understand it correctly. 

It means that the subspace topology on $U$  induced by $X$ is the same topology as the discrete topology. 
Yes, if $U$ is open in $X$ and $A$ is open in $U$ then $A$ is open in $X$. 
"I do know what a discrete space is, so as far as I understand it, the meaning is that each $A\subseteq U$ is open in the relative topology in $U$"
Yes, the subspace topology (or relative topology) means that $A \subseteq U$ is open if and only if there is an open set $O$ in $X$ such that $O \cap U = A$. 
The discrete topology is the topology where every set is open. This is induced by the discrete metric. 
Consider the set $U=\mathbb Z$ in $X=\mathbb R$. Then $U$ endowed with the subspace topology is a discrete space, in particular, every singleton $\{k\}$ is open in the subspace topology. Of course this is false for the set $\{k\} \subseteq \mathbb R$.
