Nowhere dense set

In general topology, a nowhere dense set in a topological space is a set whose closure has empty interior.

An infinite Cartesian product of non-empty non-compact spaces has the property that every compact subset is nowhere dense.

A finite union of nowhere dense sets is again nowhere dense.

A first category space or meagre space is a countable union of nowhere dense sets: any other topological space is of second category. The Baire category theorem states that a non-empty complete metric space is of second category.

References

• J.L. Kelley (1955). General topology. van Nostrand, 145,201.