Some Basic Properties Of Hyper-sober Spaces

This paper mainly discusses some basic properties of hyper-sober space.The hyper-soberity of three kinds of special spaces are investigated.Some relations between the hyper-soberity of a T0 space X and that of its Smyth power space and Hoare power space are discussed.A counterexample is given to show that the product of a countable infinite family of hyper-sober spaces is not hyper-sober in general.It is proved that if a T0 space X is not hyper-sober,then its hyper-sobrification does not exist.