Namespaces
Variants
Actions

Difference between revisions of "Relatively-open (-closed) set"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(LaTeX)
 
Line 1: Line 1:
''set open (closed) relative (or with respect to) to a certain set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081060/r0810601.png" /> in a [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081060/r0810602.png" />''
+
''set open (closed) relative (or with respect to) to a certain set $E$ in a [[topological space]] $X$"
  
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081060/r0810603.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081060/r0810604.png" /> such that
+
A set $M$ in $X$ such that
 +
$$
 +
M = E \setminus \overline{(E\setminus M)} \, \ \ (\, M = E \cap \bar M\,)
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081060/r0810605.png" /></td> </tr></table>
+
(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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081060/r0810606.png" />, it is necessary and sufficient that it is the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081060/r0810607.png" /> and a certain open (closed) set.
 
  
  
  
 
====Comments====
 
====Comments====
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081060/r0810608.png" /> in a topological space is relatively open (relatively closed) with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081060/r0810609.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081060/r08106010.png" /> is open (respectively, closed) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081060/r08106011.png" /> for the [[Relative topology|relative topology]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081060/r08106012.png" />.
+
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=16289
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article