Coarser and finer topologies
In the mathematical sub-area of topology, coarser and finer topologies are special set systems that are related to one another in a certain way. A topology is called a coarser topology than another topology if it is contained in this, and a finer topology if it contains this.
definition
Given is a set with two topologies and . Is
- ,
so the topology is called stronger or finer than . Conversely, weaker or coarser than is then mentioned .
Examples
For a given is the trivial topology
the coarsest possible topology and thus included in every further topology. This already applies due to the definition of a topology that must always contain the basic set and the empty set.
The reverse is the discrete topology
the finest topology, since it contains all subsets of the basic set by definition of the power set. There can therefore be no topology that really contains more sets than .
A nontrivial example of coarser and finer topologies are the weak topology and the standard topology on standardized spaces . The weak topology is defined as the initial topology : It is the coarsest topology on the base space , so that all linear, standardized functionals are continuous. The standard topology, however, is made up of the standard spheres
generated. The weak topology is then weaker (or coarser) than the standard topology.
properties
The following applies for two topologies and on a set : It is exactly when the identical mapping is continuous.
In metric spaces and standardized spaces , many properties are inherited from the metrics or norms on the corresponding topologies. For example, if the norm is on a stronger norm than , the norm topology induced by is finer than the norm topology induced by. The same statement also applies to the topologies generated by metrics.
In general: have finer topologies
- more open quantities
- more completed sets
- more continuous mappings into any further topological spaces
- less continuous mappings of any further topological spaces
- less compact quantities and
- less convergent consequences
Association of topologies
Is a lot, it can be naturally through inclusion a partial order to define. This semi-order structure is inherited by the set
- .
It is even more true: becomes a complete association with regard to the partial order induced by the inclusion :
To do this, one defines two topologies
- than the cut as well
- than the topology generated by,
since the union of topologies generally only provides the sub-base of a topology. Furthermore one defines for arbitrary and therefore especially infinite families
- than the cut as well
- than the topology generated by the sub-base .
As a complete association is also restricted, in this case by the discrete topology on the one hand and the indiscreet topology on the other. However, the association is not distributive.
literature
- Hans Wilhelm Alt : Linear Functional Analysis . 6th edition. Springer-Verlag, Berlin Heidelberg 2012, ISBN 978-3-642-22260-3 , doi : 10.1007 / 978-3-642-22261-0 .
- Dirk Werner : Functional Analysis . 7th, corrected and enlarged edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21016-7 , doi : 10.1007 / 978-3-642-21017-4 .
Individual evidence
- ↑ René Bartsch: General Topology. De Gruyter, 2015, ISBN 978-3-110-40618-4 , p. 79 ( limited preview in the Google book search).
- ^ HJ Kowalsky: Topological Spaces. Springer-Verlag, 2014, ISBN 978-3-034-86906-5 , p. 59 ( limited preview in Google book search).