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

${\displaystyle U_{\epsilon }(x):=\{y\in X\mid \|x-y\|<\epsilon \}}$

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

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

${\displaystyle {\mathcal {T}}:=\{\tau \mid \tau {\text{ ist Topologie auf }}X\}}$.

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

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

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).