If X Y and Y is lower bounded, then inf Y inf X . Let X be a non empty subset of and Y be a subset of . If X Y and Y is upper bounded, then sup X sup Y . Let X be a subset of . X is upper bounded and sup X X .
X is lower bounded and inf X X .
Let us note that there exists a subset of which is non empty, closed, and bounded. Let R be a family of subsets of . For every subset X of such that X R holds X is open. For every subset X of such that X R holds X is closed. Let X be a subset of .