Meagre set

In the general topology and in particular the descriptive set theory is referred to amounts to be lean (English meager or meager ) that are small or negligible in a certain sense: they are the union of countably many nowhere dense subsets of a topological space. That meager quantities actually in appropriate areas in a meaningful way can be considered small, and not the whole space is about lean, is guaranteed by the theorem of Baire.

A set whose complement is meager, is called komagere amount or residual quantity (English residual set, comeagre set or comeager set).

Alternatively, meager amounts as the subsets define the subset of a union of countably many closed sets without interior points are.

Examples

• Each countable set is lean, if singleton sets are nowhere dense.
• In particular, in each T1 - space ( each singleton set is closed ) without isolated points ( no singleton is open) every countable set lean.
• A meager amount does not contain isolated points of the surrounding space, because such would contribute to the interior of the lot.
• Each dense open set and every countable intersection of dense open sets are residually. Because the complement of a dense open set is nowhere dense: Otherwise it would have as a closed set not empty interior, which lies outside the given open set, which thus could not be tight.
• For instance, the set of rational numbers lean in the set of real numbers.
• Accordingly, the set of irrational numbers is residually.
• The set of all positive real numbers is not lean, but not residually as the complement is not too lean.
• Each nowhere dense set is lean, such as the Cantor set.
• Skinny quantities are closed under countable union.