Coarser topology

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 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 ${\tau }_1$.