Meagre subsetes of [0,1] A subset $A$ of a topological space $X$ is **nowhere dense** if it's closure has empty interior. It is true that every closed nowhere dense set is nowhere dense. A set is **meagre** if it is a countable union of nowhere dense sets. What is an example of a closed meagre subset of $[0,1]$ which fails to be nowhere dense?

Consider the three properties: closed, meager, not nowhere dense. There are sets with any two of these properties but no sets with all three. Examples:

* Closed+meager: A finite set; a Cantor set.
* Closed+not nowhere dense: a closed interval.
* Meager+not nowhere dense: The rationals; a union of fat Cantor sets with full measure.



We cannot have all three properties, because:

1. If a set is meager, then by the Baire category theorem it has empty interior.
2. If a set is not nowhere dense, then its closure has nonempty interior.
3. If a set is closed, then it is equal to its closure.



So having all three properties gives a contradiction (provided the underlying space is a complete metric space).