View source for Relatively-open (-closed) set

''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" />''

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

<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 <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====
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" />.

====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>