This article is about a basic definition in topology.
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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 .
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: