# Finer 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 **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 .

## Related notions

### Universal constructions

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.

In contrast, *coarsest* topologies arise in pullbacks, for instance, in the subspace topology.

### 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:

Category:Refining-preserved properties of topological spaces