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.
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