Coarser topology

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 $$\tau_1$$.

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