TITLE: Show that $D =\{ x + y \mid x \in (0,1) ,y \in [1,2) \}$ is open or closed
QUESTION [2 upvotes]: I have $\mathbb{R}$ with the euclidian metric $|x-y|$ for $x,y\in \mathbb{R}$. I want to show with arguments that the set $D =\{ x + y \mid x \in (0,1) ,y \in [1,2) \}$ is open or closed.
As a definition for open/closed I have:
Definition of open:
Let $(X,d)$ be a metric space and $U \subseteq X$. The set $U$ is called open set if: for every $U$ there exists some $\epsilon > 0 $ such that $B_\epsilon (x) \subseteq U$.
Definition of closed:
Let $(X,d)$ be a metric space and $F \subseteq X$. The set $F$ is called closed if $X\setminus F$ is called open.
Actually I do not have a real idea how to start yet. First I would draw the graph, but here I do not know where to start...
I would really appreciate your answers!!!

REPLY [1 votes]: Alternatively, show that if $U$ is open and $X$ is any set then $U+X$ is open.
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