Dense Sets

A subsetof a setis dense in X if the closure ofis equal toie

With this definition the following statements are equivalent.

1.is dense in

2. Ifis a closed set,is dense inandthen

3. Each nonempty open set incontains an element of

The complementofhas empty interior.

Proof

1. Ifis dense inthenSincebutsinceis closed hencetherefore

2. Letbe an open set withThencontradicting 2 becauseis closed.

3. Supposeis open therefore there is a nonempty setbuthenceandcontains no points ofHencecontains some element of

4.hence