# 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: