Coarser and finer topologies

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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.


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 .


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.


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

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.


Individual evidence

  1. René Bartsch: General Topology. De Gruyter, 2015, ISBN 978-3-110-40618-4 , p. 79 ( limited preview in the Google book search).
  2. ^ HJ Kowalsky: Topological Spaces. Springer-Verlag, 2014, ISBN 978-3-034-86906-5 , p. 59 ( limited preview in Google book search).