Abstract. It is shown that the 5-closed spaces introduced by Travis Thompson using semiopen sets may be characterized as spaces where covers by regular closed sets have finite subcovers. S-closed is contagious and semiregular but is neither productive nor preserved by continuous surjections. Extremally disconnected QHC spaces are 5-closed and maximal S-closed spaces are precisely the maximal QHC spaces which are extremally disconnected. Definition 1. A topological space (X, t) is quasi-H-closed (denoted QHC) if every open cover has a finite proximate subcover (every open cover has a finite subfamily whose closures cover the space). In Hausdorff spaces, QHC is 77-closed. Definition 2. A set A in a topological space (A', t) is semiopen if intT A c A c clT intT A where intT A denotes the interior of A with respect to the topology t and clT A is the closure of A with respect to the topology t.Definition 3. A topological space (X, t) is S-closed if every semiopen cover has a finite proximate subcover.It is obvious that every 5-closed space is QHC but the converse is not true [7]. It has been shown that a space is QHC if and only if every cover of regular open sets has a finite proximate subcover. A similar result holds for 5-closed spaces. Proof. If the space is 5-closed then the result follows. If the space is not 5-closed then there is a semiopen cover {Aß\ ß E 23} which has no finite proximate subcover. Then {(intT clT Aß) u Aß\ ß E 93} is a regular semiopen cover which has no finite proximate subcover since Ap c (intT clT Ap) u^c clT Aß for each ß E 33. Q.E.D.
CorollaryI. An extremally disconnected QHC space is S-closed.
Abstract. A topological space (X, T) with property R is maximal R (minimal R) if T is a maximal (minimal) element in the set R(X) of all topologies on the set X having property R with the partial ordering of set inclusions. The properties of maximal topologies for compactness, countable compactness, sequential compactness, Bolzano-Weierstrass compactness, and Lindelöf are investigated and the relations between these spaces are investigated. The question of whether any space having one of these properties has a strictly stronger maximal topology is investigated. Some interesting product theorems are discussed. The properties of minimal topologies and their relationships are discussed for the quasi-i7, Hausdorff quasi-P, and P topologies.I. Background and introduction.For a given topological property R and a set X, we let R(A') denote the set of all topologies on X which have property R and observe that R(A) is partially ordered by set inclusion. A topological space (X, T) is maximal R (R-maximal) provided that F is a maximal element in R(A'). A topological space (A, T) is minimal R (R-minimal) if F is a minimal element in R(A"). A topology T' on the set A'is finer than a topology 7" if F 3 T; the topology F is said to be coarser than the topology T'.
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