Relatively-open (-closed) set

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)