Finer topology

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This article is about a basic definition in topology.
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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 X be a set and \tau_1 and \tau_2 be two topologies on X. We say that \tau_1 is finer than \tau_2 if the following equivalent conditions are satisfied:

  • Any open set for \tau_2 is also open for \tau_1
  • Any closed set for \tau_2 is also closed for \tau_1
  • The identity map (X,\tau_1) \to (X,\tau_2) is a continuous map

The opposite notion is that of coarser topology. In this case, \tau_2 is coarser than \tau_1.

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