Non Decreasing Sequences of Closed Sets and Non Increasing Sequences of Open Sets

Theorem

Ifis closed and is a union of a countable number of closed sets then there is a non decreasing sequence of closed setswhere

Ifis closed and is a intersection of a countable number of open sets then there is a non increasing sequence of open sets where

Proof

Sinceis closed and is a union of a countable number of closed setswith eachclosed.

Let

Then the setsare closed andand

Similarly ifis closed and is a intersection of a countable number of open sets thewhereare open sets.

Let