TITLE: Is this set $\{(x,y) \in \mathbb R^2 : |x|+|y|\leq 1\}$ compact?
QUESTION [7 upvotes]: Is this set $\{(x,y) \in \mathbb R^2 : |x|+|y|\leq 1\}$ compact?
I know that is closed and bounded so compact but I don't know how to show it is closed and bounded mathematically.
This is the graph of $|x|+|y|\leq 1$

REPLY [5 votes]: It is bounded since it's contained in the ball of radius $2$. It's closed because it's the preimage of the closed set $[0,1]\subseteq \mathbb R$ via the continuous function $(x,y)\mapsto |x|+|y|$