Topological Space: Difference between revisions

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imported>Aleksander Stos
(short time editor-aimed message before deletion -- please do not develop this article but help moving useful content to the right place.)
imported>Jitse Niesen
(replace by #REDIRECT Topological space which now contains all the information in this article and more)
 
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;It was suggested that this article is merged with [[Topological space]], the latter being the right title according to our naming conventions.
#REDIRECT [[Topological space]]
 
 
Topological spaces are
 
==The Open and Closed Set Axioms==
 
Let <math>X</math> be a set, and <math>\tau</math> a collection of subsets of <math>X</math> (which will be called the ''open subsets'' of <math>X</math> with respect to the topology <math>\tau</math>) verifying the following axioms:
#<math>\varnothing\in\tau</math>
#<math>X\in\tau</math>
#Any finite intersection of sets in <math>\tau</math> is again in <math>\tau</math>; i.e., if <math>U_1,\ldots,U_n\in\tau</math>, then <math>U_1\cap\ldots\cap U_n\in\tau</math>.
#Any union of a family of sets <math>\{U_{\lambda}\}_{\lambda\in\Lambda}\subseteq\tau</math> is in <math>\tau</math>; i.e., <math>\bigcup_{\lambda\in\Lambda} U_{\lambda}\in\tau</math>.
When these axioms are satisfied, we say that <math>(X,\tau)</math> is a topological space of open sets <math>\tau</math>.
 
 
==The Neighborhood Axioms==
 
One can phrase a set of axioms for the definition of a topological space by defining the ''neighborhoods'' of points in that space.  This is particularly useful when one considers topologies on topological abelian groups and topological rings by subgroups or ideals, respectively, because knowing the neighborhoods of any point is equivalent to knowing the neighborhoods of <math>0</math>.
 
 
 
 
 
==Examples==
 
# [[Metric space]]s
 
==The Category of Topological Spaces==
 
 
 
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Latest revision as of 08:46, 5 December 2007

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