Finer topology

From Topospaces
Jump to: navigation, search
This article is about a basic definition in topology.
VIEW: Definitions built on this | Facts about this | Survey articles about this
View a complete list of basic definitions in topology

Definition

Symbol-free definition

Given two topologies on a set, one is said to be finer than the other if the following equivalent conditions are satisfied:

  • Every set that is open as per the second topology, is also open as per the first
  • Every set that is closed as per the second topology, is also closed as per the first
  • The identity map is a continuous map from the first topology to the second

Definition with symbols

Let be a set and and be two topologies on . We say that is finer than if the following equivalent conditions are satisfied:

  • Any open set for is also open for
  • Any closed set for is also closed for
  • The identity map is a continuous map

The opposite notion is that of coarser topology. In this case, is coarser than .

Related notions

Universal constructions

The finest possible topology on a set is the discrete topology, where all subsets are deemed open (and hence, also closed). There ar esituations where we want to impose on a topological space the finest topology subject to certain constraints. An example is the quotient topology, which is the finest possible topology on a quotient space to make a set-theoretic quotient map continuous.

A more general example is that of the topology on a pushout.

In contrast, coarsest topologies arise in pullbacks, for instance, in the subspace topology.

Effect on topological space properties

Moving from a particular topology on a set to a finer topology might have various kinds of effect on topological space properties. A list of properties of topological spaces that are preserved upon passing to finer topologies, is available at:

Category:Refining-preserved properties of topological spaces