| This paper answers a problem about the heredity ofσ? orthospact space and deduces that there are two groups of equivalent characterizations for all such X from this result, and study some properties of hyperspaces C ( D, X ),we have thefollowing conclusions: 1. X is heredity ofσ? orthospact space if and only if every scattered partition of X is aσ? interior preserving of open expansions.2. Let X be a topological space, then the following conditions are equivalent:(1) X is heredity ofσ? orthospact space;(2)Every monotone decreasing family { Fα:α<γ} of close sets of X has aσ?interior preserving open family V =∪n∈ωV n such that,for every ?α<γ, X ? Fα=∪{V∈V :V∩Fα=?}.(3)Every monotone increasing family U = {Uα:α<γ} of open sets of X has aσ?interior preserving open refinement V =∪n∈ωVn such that,for every ?α<γ, Uα=∪{V∈V :V ?Uα}.3.Let X and Y be continua.A mapping f :X→Y is confluent if and only if for any D∈C ( X),∧f (C ( D , X )) = C ( f ( D ), Y).4. Let X and Y be continua and let h :X→Y be a homeomorphism. Then C ( D , X )≈C ( h ( D ), Y).5.Let X be a continuum. If D∈C ( X) is such that for any A, B∈C ( D , X), A and B are comparable, then C ( D , X ) is an arc.6. Let X be a continuum and let D∈C ( X). Then neither D nor X is a cut set of C ( D , X ).7. Let X be a continuum and let D∈C ( X).Suppose A∈C ( D , X) is such that either D A X . Then A is terminal at D if and only if A is a cut set of C ( D , X ). 8.Ifαis an order arc in 2X beginning with A0∈C ( X),thenα? C ( X).9.Let D be a proper subcontinuum of a continuum X .Then, D is terminal in X if and only if C ( D , X ) = { K∈C ( X ),D ? K}is an order arc.10.Let X be a continuum and let n∈N be such that the set { D∈C ( X ), C ( D , X) has cut set } is at most countable and for each D∈C ( X),dim(C ( D , X ))< N.Then every proper and nondegenerate subcontinuum of X is decomposable.11.The following conditions are equivalent for a continuum X : (1) X is hereditarily indecomposable. (2) C ( D , X )is an arc for each D∈C ( X).
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