Coarser topology

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