Finer topology

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 $$X$$ be a set and $$\tau_1$$ and $$\tau_2$$ be two topologies on $$X$$. We say that $$\tau_1$$ is finer than $$\tau_2$$ if the following equivalent conditions are satisfied:


 * Any open set for $$\tau_2$$ is also open for $$\tau_1$$
 * Any closed set for $$\tau_2$$ is also closed for $$\tau_1$$
 * The identity map $$(X,\tau_1) \to (X,\tau_2)$$ is a continuous map

The opposite notion is that of coarser topology. In this case, $$\tau_2$$ is coarser than $$\tau_1$$.

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