Finer topology

This article is about a basic definition in topology.
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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:

Category:Refining-preserved properties of topological spaces