Difference between revisions of "Relatively-open (-closed) set"
From Encyclopedia of Mathematics
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| − | ''set open (closed) relative (or with respect to) to a certain set | + | ''set open (closed) relative (or with respect to) to a certain set $E$ in a [[topological space]] $X$" |
| − | A set | + | A set $M$ in $X$ such that |
| + | $$ | ||
| + | M = E \setminus \overline{(E\setminus M)} \, \ \ (\, M = E \cap \bar M\,) | ||
| + | $$ | ||
| − | + | (the bar denotes the operation of closure, cf. [[Closure of a set|Closure of a set]]). For a certain set to be open (closed) relative to $E$, it is necessary and sufficient that it is the intersection of $E$ and a certain open (closed) set. | |
| − | |||
| − | (the bar denotes the operation of closure, cf. [[Closure of a set|Closure of a set]]). For a certain set to be open (closed) relative to | ||
====Comments==== | ====Comments==== | ||
| − | A set | + | A set $M$ in a topological space is relatively open (relatively closed) with respect to $E$ if and only if $M \cap E$ is open (respectively, closed) in $E$ for the [[relative topology]] on $E$. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.S. [P.S. Aleksandrov] Alexandroff, H. Hopf, "Topologie" , Chelsea, reprint (1972) pp. 33ff, 44ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> C. Kuratowski, "Introduction to set theory and topology" , Pergamon (1961) pp. 128ff (Translated from French)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> P.S. [P.S. Aleksandrov] Alexandroff, H. Hopf, "Topologie" , Chelsea, reprint (1972) pp. 33ff, 44ff</TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> C. Kuratowski, "Introduction to set theory and topology" , Pergamon (1961) pp. 128ff (Translated from French)</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | {{TEX|done}} | ||
| + | |||
| + | [[Category:General topology]] | ||
Latest revision as of 13:57, 9 November 2014
set open (closed) relative (or with respect to) to a certain set $E$ in a topological space $X$"
A set $M$ in $X$ such that $$ M = E \setminus \overline{(E\setminus M)} \, \ \ (\, M = E \cap \bar M\,) $$
(the bar denotes the operation of closure, cf. Closure of a set). For a certain set to be open (closed) relative to $E$, it is necessary and sufficient that it is the intersection of $E$ and a certain open (closed) set.
Comments
A set $M$ in a topological space is relatively open (relatively closed) with respect to $E$ if and only if $M \cap E$ is open (respectively, closed) in $E$ for the relative topology on $E$.
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
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