This article is about a basic definition in topology.
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Given two topologies on a set, one is said to be finer than the other if the following equivalent conditions are satisfied:
- Every set that is open as per the second topology, is also open as per the first
- Every set that is closed as per the second topology, is also closed as per the first
- The identity map is a continuous map from the first topology to the second
Definition with symbols
Let be a set and and be two topologies on . We say that is finer 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 coarser topology. In this case, is coarser than .
The finest possible topology on a set is the discrete topology, where all subsets are deemed open (and hence, also closed). There ar esituations where we want to impose on a topological space the finest topology subject to certain constraints. An example is the quotient topology, which is the finest possible topology on a quotient space to make a set-theoretic quotient map continuous.
A more general example is that of the topology on a pushout.
Effect on topological space properties
Moving from a particular topology on a set to a finer topology might have various kinds of effect on topological space properties. A list of properties of topological spaces that are preserved upon passing to finer topologies, is available at: