Coarser topology: Difference between revisions

Revision as of 17:38, 11 December 2007

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 $τ_{1}$ .

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 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 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>.