Show that $(0,1]$ is open in $[0,1]$ but $(0,1]$ is not open in $\mathbb{R}$

Question

I am very new to Topology, so excuse my lack knowledge;

Show that $A=(0,1]$ is open in $[0,1]$:

I am suppose to choose $x$ in $(0,1]$. And construct a ball around with radius $r\leq\min\{ \lvert x-1 \lvert,x\}$? Should I show that the complement is closed?

I could also use a hint on showing that $(0,1]$ is not open i $\mathbb{R}$.

Answer 1

In the subspace topology on $[0,1]$ $(0,1]$ is open because $$ (0,1]=(0,1)\cap[0,1] $$ However in $\mathbb{R}$, $[0,1]$ is not open because it is not a open set in $\mathbb{R}$.

Answer 2

$ (0;1] $ is not open in R as you cannot find an origin O and an $\epsilon > 0 $ so that B$(O,\epsilon)$ contains the set $(0;1]$