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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 $\tau$ of a topological space $(X, \tau)$ be defined as a system $\tau$ of subsets of the set $X$ such that:  
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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:  
 
# $X\in\tau$, $\emptyset\in\tau$;  
 
# $X\in\tau$, $\emptyset\in\tau$;  
 
# if $O_i\in\tau$, where $i=1,2$, then $O_1\cap O_2\in\tau$;  
 
# if $O_i\in\tau$, where $i=1,2$, then $O_1\cap O_2\in\tau$;  
# if $O_{\alpha}\in\tau$, where $\alpha\in\mathfrak{A}$, then $\bigcup\{O_{\alpha} : \alpha\in\mathfrak{A} \}$.  
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# if $O_{\alpha}\in\tau$, where $\alpha\in\mathfrak{A}$, then $\bigcup\{O_{\alpha} : \alpha\in\mathfrak{A} \} \in \tau$.  
  
 
The open sets in the space $(X, \tau)$ are then the elements of the topology $\tau$ and only them.
 
The open sets in the space $(X, \tau)$ are then the elements of the topology $\tau$ and only them.
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR>
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</table>

Latest revision as of 20:40, 9 April 2016

2020 Mathematics Subject Classification: Primary: 54A05 [MSN][ZBL]

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 $\bigcup\{O_{\alpha} : \alpha\in\mathfrak{A} \} \in \tau$.

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://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