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# Dense set and Nowhere dense set ... Watch

1. Hi ,

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).

2. 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).

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Updated: December 12, 2010
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