Coarser 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 coarser than the other if the following equivalent conditions are satisfied:

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

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 coarser than \tau_2 if the following equivalent conditions are satisfied:

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

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

Related notions

Universal constructions

The trivial topology (the topology where the only open subsets are the whole space and the empty set) is the coarsest possible topology on a set. We are often interested in the coarsest possible topology on a set subject to additional conditions. For instance, the subspace topology is the coarsest topology on a subset to make the inclusion map continuous. More generally, pullbacks are given the coarsest possible topology to make the maps from them continuous.

Effect on topological space properties

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

Category:Coarsening-preserved properties of topological spaces