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Difference between revisions of "Open set"

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''in a topological space''
 
''in a topological space''
  
An element of the topology (cf. [[Topological structure (topology)|Topological structure (topology)]]) of this space. More specifically, let the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068310/o0683101.png" /> of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068310/o0683102.png" /> be defined as a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068310/o0683103.png" /> of subsets of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068310/o0683104.png" /> such that: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068310/o0683105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068310/o0683106.png" />; 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068310/o0683107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068310/o0683108.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068310/o0683109.png" />; and 3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068310/o06831010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068310/o06831011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068310/o06831012.png" />. The open sets in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068310/o06831013.png" /> are then the elements of the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068310/o06831014.png" /> and only them.
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An element of the topology (cf. [[Topological structure (topology)|Topological structure (topology)]]) of this space. More specifically, let the topology $\tau$ of a topological space $(X, \tau)$ be defined as a system $\tau$ of subsets of the set $X$ such that:  
 
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# $X\in\tau$, $\emptyset\in\tau$;  
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# if $O_i\in\tau$, where $i=1,2$, then $O_1\cap O_2\in\tau$;  
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# if $O_{\alpha}\in\tau$, where $\alpha\in\mathfrak{A}$, then $\bigcup\{O_{\alpha} : \alpha\in\mathfrak{A} \}$.  
  
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The open sets in the space $(X, \tau)$ are then the elements of the topology $\tau$ and only them.
  
====Comments====
 
  
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR></table>

Revision as of 05:04, 22 April 2013


in a topological space

An element of the topology (cf. Topological structure (topology)) of this space. More specifically, let the topology $\tau$ of a topological space $(X, \tau)$ be defined as a system $\tau$ of subsets of the set $X$ such that:

  1. $X\in\tau$, $\emptyset\in\tau$;
  2. if $O_i\in\tau$, where $i=1,2$, then $O_1\cap O_2\in\tau$;
  3. if $O_{\alpha}\in\tau$, where $\alpha\in\mathfrak{A}$, then $\bigcup\{O_{\alpha} : \alpha\in\mathfrak{A} \}$.

The open sets in the space $(X, \tau)$ are then the elements of the topology $\tau$ and only them.


References

[a1] R. Engelking, "General topology" , Heldermann (1989)
How to Cite This Entry:
Open set. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Open_set&oldid=29690
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article