Difference between revisions of "Open set"
From Encyclopedia of Mathematics
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| − | {{TEX|done}} | + | {{TEX|done}}{{MSC|54A05}} |
''in a topological space'' | ''in a topological space'' | ||
| − | An element of the topology (cf. [[ | + | 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 $\bigcup\{O_{\alpha} : \alpha\in\mathfrak{A} \}$. | + | # 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> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Engelking, "General topology" , Heldermann (1989)</TD></TR> | ||
| + | </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:
- $X\in\tau$, $\emptyset\in\tau$;
- if $O_i\in\tau$, where $i=1,2$, then $O_1\cap O_2\in\tau$;
- if $O_{\alpha}\in\tau$, 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
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