# 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 ${\ displaystyle X}$${\ displaystyle \ tau _ {1}}$${\ displaystyle \ tau _ {2}}$

${\ displaystyle \ tau _ {1} \ subset \ tau _ {2}}$,

so the topology is called stronger or finer than . Conversely, weaker or coarser than is then mentioned . ${\ displaystyle \ tau _ {2}}$ ${\ displaystyle \ tau _ {1}}$${\ displaystyle \ tau _ {1}}$ ${\ displaystyle \ tau _ {2}}$

## Examples

For a given is the trivial topology${\ displaystyle X}$

${\ displaystyle \ tau _ {1}: = \ {X, \ emptyset \}}$

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

${\ displaystyle \ tau _ {2}: = {\ mathcal {P}} (X)}$

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 . ${\ displaystyle \ tau _ {2}}$

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 ${\ displaystyle X}$${\ displaystyle X}$

${\ displaystyle U _ {\ epsilon} (x): = \ {y \ in X \ mid \ | xy \ | <\ epsilon \}}$

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. ${\ displaystyle \ tau _ {1}}$${\ displaystyle \ tau _ {2}}$${\ displaystyle X}$${\ displaystyle \ tau _ {2} \ subseteq \ tau _ {1}}$ ${\ displaystyle \ operatorname {id} _ {X} \ colon (X, \ tau _ {1}) \ to (X, \ tau _ {2}), x \ mapsto x}$

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. ${\ displaystyle \ | \ cdot \ | _ {1}}$${\ displaystyle X}$${\ displaystyle \ | \ cdot \ | _ {2}}$${\ displaystyle \ | \ cdot \ | _ {1}}$${\ displaystyle \ | \ cdot \ | _ {2}}$

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 ${\ displaystyle X}$${\ displaystyle {\ mathcal {P}} ({\ mathcal {P}} (X))}$

${\ displaystyle {\ mathcal {T}}: = \ {\ tau \ mid \ tau {\ text {is topology on}} X \}}$.

It is even more true: becomes a complete association with regard to the partial order induced by the inclusion : ${\ displaystyle {\ mathcal {T}}}$

To do this, one defines two topologies ${\ displaystyle \ tau, \ sigma \ in {\ mathcal {T}}}$

${\ displaystyle \ tau \ wedge \ sigma}$than the cut as well${\ displaystyle \ tau \ cap \ sigma}$
${\ displaystyle \ tau \ vee \ sigma}$than the topology generated by,${\ displaystyle \ tau \ cup \ sigma}$

since the union of topologies generally only provides the sub-base of a topology. Furthermore one defines for arbitrary and therefore especially infinite families${\ displaystyle (\ tau _ {i}) _ {i \ in I} \ subseteq {\ mathcal {T}}}$

${\ displaystyle \ bigwedge _ {i \ in I} \ tau _ {i}}$than the cut as well${\ displaystyle \ bigcap _ {i \ in I} \ tau _ {i}}$
${\ displaystyle \ bigvee _ {i \ in I} \ tau _ {i}}$than the topology generated by the sub-base .${\ displaystyle \ bigcup _ {i \ in I} \ tau _ {i}}$

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. ${\ displaystyle {\ mathcal {T}}}$${\ displaystyle {\ mathcal {T}}}$

## 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).