# Nowhere dense set

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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.