Namespaces
Variants
Actions

Relatively-open (-closed) set

From Encyclopedia of Mathematics
Revision as of 17:16, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

set open (closed) relative (or with respect to) to a certain set in a topological space

A set in such that

(the bar denotes the operation of closure, cf. Closure of a set). For a certain set to be open (closed) relative to , it is necessary and sufficient that it is the intersection of and a certain open (closed) set.


Comments

A set in a topological space is relatively open (relatively closed) with respect to if and only if is open (respectively, closed) in for the relative topology on .

References

[a1] P.S. [P.S. Aleksandrov] Alexandroff, H. Hopf, "Topologie" , Chelsea, reprint (1972) pp. 33ff, 44ff
[a2] C. Kuratowski, "Introduction to set theory and topology" , Pergamon (1961) pp. 128ff (Translated from French)
How to Cite This Entry:
Relatively-open (-closed) set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Relatively-open_(-closed)_set&oldid=34421
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article