Proof That the Set of All Open Intervals is a Topology on the Set of Real Numbers

Letbe the set of all real numbers. A setis open if for eachthere exists such that

is a metric space whereLet the family of all open intervals inbe calledthenis a topology on

Proof

Obviouslyandare open insoand

Supposeare open sets, then

In factsuch thatwhere

Suppose eachis open thenis also open (may be infinite here), since