Some Properties And Structures Of Directed Closure Spaces

A topological space X is called a directed closure space(DC space for short)if X is T0 and any irreducible closed set of X is the closure of a directed set of X(endowed with the specialization order).In this paper,some basic properties of directed closure spaces are discussed.The main results are:(1)DC spaces are hereditary with respect to open subspace;(2)A counterexample is given to show that DC spaces are not hereditary with respect to saturated subspaces in general;(3)A poset endowed with the Alexandroff topology and its Smyth power space are DC spaces;(4)A poset endowed with the upper topology is quasisober if and only if it is a DC space;(5)There is a space which is not a DC space but its Smyth power space is a DC space;(6)DC-reflection of ?J not existent,where J is the Johnstone dcpo.