부산대학교 도서관

This continues a series of papers that study a topology, defined by E. Čech in 1937 [Časopis Pěst. Mat. Fys. {\bf 66} (1937), 225--264], with weaker axioms than the usual ones: a topological space is a set $P$ with a closure operator $u$ satisfying: (1) $u\phi=\phi$, (2) $M\subset uM$, (3) $M_1\subset M_2$ implies $uM_1\subset uM_2$. $M$ is closed if and only if $uM=M$, open if and only if $P-M$ is closed. $M^\circ$, the interior of $M$, is $P-u(P-M)$, and $O$ is a neighborhood of $M$ if and only if $M\subset O^\circ$. \par Define $\scr O_u(M)$ to be the set of all neighborhoods of $M,s_uM=\bigcap\scr O_u(M)$, and $i_uM=\bigcap\{0^\circ|O\in\scr O_u(M)\}$. $i_uM$ is called the inner constellation of $M$, and since the interior of a set may fail to be open, $i_uM\subset s_uM$ may be a proper inclusion. \par The present paper studies various properties $(f)$ that a topological space $(P,u)$ may have, especially investigating property $(\alpha)$, that $i_uM=s_uM$. For example, let $u_f$ be the supremum of all topologies contained in $u$ that have property $(f)$, and let $u^f$ be the infimum of all topologies containing $u$ that have property $(f)$, if these exist. The authors show that $u^\alpha$ always exists and $u_\alpha$ exists if and only if $u$ itself has property $(\alpha)$. ($u_f$ and $u^f$ are called the lower and upper modifications of $u$, respectively.)