Lean amount
A lean set , also called a set (of) the first (Baire) category , is a set in set- theoretical topology that has a small extent from a topological point of view. A quantity that is not lean is also called a fat quantity or a quantity (of) the second (baire) category . In contrast, the complement of a lean set is called a lean set or a residual set .
These terms are used, for example, in the formulation of Baire's category set , which states that complete metric spaces are "topologically large", as well as in the abstraction of this property using Baire spaces .
It should be noted that, contrary to the naming as a set of first / second category, there is no direct reference to category theory .
definition
A topological space is given . A set is called lean or of the first (Baire) category if it is the countable union of nowhere dense sets . A crowd does n't mean tight anywhere if the inside of its closure is empty.
Constructive terms
A set is called a lean or residual set when it is the complement of a lean set.
A quantity that is not lean is called fat or of the second (baire) category.
Examples
- Any countable set is lean if one-element sets are nowhere dense.
- In particular, in any T 1 -space (every singleton set is closed) without isolated points (no singleton set is open), every countable set is lean.
- A meager set does not contain isolated points of the surrounding space, for these would contribute to the interior of the set.
- Every dense open set and every countable intersection of dense open sets are residual. Because the complement of a dense open set is nowhere dense: Otherwise, as a closed set, it would have a non-empty interior that would lie outside the given open set, which therefore could not be dense.
- For example, the set of rational numbers is lean in the set of real numbers .
- Accordingly, the set of irrational numbers is residual.
- The set of all positive real numbers is not lean, but neither is it residual, since the complement is not lean either.
- Any nowhere dense crowd is lean, such as the Cantor crowd .
- Lean sets are closed under countable union.
See also
- The set of Baire states that in each fully metrizable space and in any locally compact Hausdorff space is densely any residual amount.
- Property of Baire
- Zero set , negligible sets in terms of measure theory .
Web links
- Category of a set . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
literature
- Boto von Querenburg : Set theoretical topology . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2001, ISBN 978-3-540-67790-1 , doi : 10.1007 / 978-3-642-56860-2 .