# Finer topology

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 $X$ 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 $\tau_1$.

## 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: