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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