# Coarser topology

This article is about a basic definition in topology.VIEW: Definitions built on this | Facts about this | Survey articles about this

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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 be a set and and be two topologies on . We say that is **coarser** 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 finer topology. In this case, is finer than .

## 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