Topology - Open and closed sets in two different metric spaces I am currently working through some topology problems, and I would like to confirm my results just for some peace of mind!
Statement: Let $(X, d_x)$ be a metric space, and $Y\subset X$ a non-empty subset. Define a metric $d_y$ on $Y$ by restriction. Then, each subset $A\subset Y $ can be viewed as a subset of $X$.

Q 1: If $A$ is open in $X$, $A$ open in $Y$.
This I found TRUE. Roughly speaking, given the preliminary conditions, $A$ cannot "lose" any of its limit points when we assess it closure in $Y$.

Q 4: If $A$ is closed in $Y$, then $A$ is closed in $X$.