We say that Y is dense in a metric space X, if the closure of Y is equal to X.
When we define the notion of "nowhere dense", wouldn't it be more natural to say:
Y is nowhere dense if its interior is equal to the empty set ?
(instead of taking the interior of the closure).
Somone please explain
Dense set and Nowhere dense set ... Watch
- Thread Starter
- 11-12-2010 22:11
- 12-12-2010 01:51
Well, (usual topologies) has empty interior, but its closure has the whole of as its interior.
I don't know why the definition uses interior-of-closure rather than just interior. Perhaps it was to avoid the somewhat bizarre situation of certain sets being simultaneously dense and nowhere dense (as above).