Coarser topology: Difference between revisions

Latest revision as of 19:40, 11 May 2008

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 coarser than the other if the following equivalent conditions are satisfied:

Every set that is open as per the first topology, is also open as per the second
Every set that is closed as per the first topology, is also closed as per the second
The identity map is a continuous map from the second topology to the first

Definition with symbols

Let $X$ be a set and $τ_{1}$ and $τ_{2}$ be two topologies on $X$ . We say that $τ_{1}$ is coarser than $τ_{2}$ if the following equivalent conditions are satisfied:

Any open set for $τ_{1}$ is also open for $τ_{2}$
Any closed set for $τ_{1}$ is also closed for $τ_{2}$
The identity map $(X, τ_{2}) \to (X, τ_{1})$ is a continuous map

The opposite notion is that of finer topology. In this case, $τ_{2}$ is finer than $τ_{1}$ .

Related notions

Universal constructions

The trivial topology (the topology where the only open subsets are the whole space and the empty set) is the coarsest possible topology on a set. We are often interested in the coarsest possible topology on a set subject to additional conditions. For instance, the subspace topology is the coarsest topology on a subset to make the inclusion map continuous. More generally, pullbacks are given the coarsest possible topology to make the maps from them continuous.

Effect on topological space properties

Moving from a particular topology on a set to a coarser topology might have various kinds of effect on topological space properties. A list of topological space properties that are preserved on passing to coarser topologies is available at:

Category:Coarsening-preserved properties of topological spaces

@@ Line 2: / Line 2: @@
 ==Definition==
+===Symbol-free definition===
+Given two topologies on a set, one is said to be '''coarser''' than the other if the following equivalent conditions are satisfied:
+* Every set that is open as per the first topology, is also open as per the second
+* Every set that is closed as per the first topology, is also closed as per the second
+* The identity map is a [[continuous map]] from the second topology to the first
+===Definition with symbols===
+Let <math>X</math> be a set and <math>\tau_1</math> and <math>\tau_2</math> be two topologies on <math>X</math>. We say that <math>\tau_1</math> is '''coarser''' than <math>\tau_2</math> if the following equivalent conditions are satisfied:
+* Any open set for <math>\tau_1</math> is also open for <math>\tau_2</math>
+* Any closed set for <math>\tau_1</math> is also closed for <math>\tau_2</math>
+* The identity map <math>(X,\tau_2) \to (X,\tau_1)</math> is a [[continuous map]]
+The opposite notion is that of [[finer topology]]. In this case, <math>\tau_2</math> is finer than <math>\tau_1</math>.
+==Related notions==
+===Universal constructions===
+The [[trivial topology]] (the topology where the only open subsets are the whole space and the empty set) is the coarsest possible topology on a set. We are often interested in the coarsest possible topology on a set subject to additional conditions. For instance, the [[subspace topology]] is the coarsest topology on a subset to make the inclusion map continuous. More generally, pullbacks are given the coarsest possible topology to make the maps ''from'' them continuous.
+===Effect on topological space properties===
+Moving from a particular topology on a set to a coarser topology might have various kinds of effect on topological space properties. A list of topological space properties that are preserved on passing to coarser topologies is available at:
+[[:Category:Coarsening-preserved properties of topological spaces]]