Coarser topology - Revision history

Vipul: 5 revisions

2008-05-11T19:40:48Z

5 revisions

← Older revision	Revision as of 19:40, 11 May 2008
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2007-12-11T17:49:19Z

← Older revision		Revision as of 17:49, 11 December 2007
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	The opposite notion is that of [[finer topology]]. In this case, <math>\tau_2</math> is finer than <math>\tau_1</math>.		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]]

2007-12-11T17:38:44Z

← Older revision		Revision as of 17:38, 11 December 2007
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	===Symbol-free 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:		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 open as per the first topology, is also open as per the second
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	===Definition with symbols===		===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:		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>

2007-12-11T17:38:22Z

Definition with symbols

← Older revision		Revision as of 17:38, 11 December 2007
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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:		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>		* 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 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>.		The opposite notion is that of [[finer topology]]. In this case, <math>\tau_2</math> is finer than <math>\tau_1</math>.

2007-08-20T09:41:27Z

← Older revision		Revision as of 09:41, 20 August 2007
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	==Definition==		==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>.

2007-08-20T09:33:57Z

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{{basicdef}}

==Definition==