The Study On Weakly Semi-Continutity And Weakly δ-Continutity On The Hyperspace

Content of abstract: This paper researched the properties of weakly semicontinuityand weaklyδ- continuity on the hyperspace and has gained the following results.1 (1)Let (X , T)and(Y ,ψ) be topological spaces , if X→P₀ is a setvaluedo ?mapping, then the following conditions are equivalent:(i) The mapping f is weakly lower semicontinuous?(ii) For every ,there have(iii) For every there have(2)Let (X,T)and(Y,ψ) be topological spaces , if f:X→P₀(Y) is a setvaluedmapping, then the following conditions are equivalent:(i) The mapping f is weakly lower semicontinuous?(ii) For every A P (Y ), there have(iii) For every B P (Y ) , there have(3) Let (X , ?)and(Y , ?) be topological spaces,if f:X→P₀(Y) is a setvalued mapping, then the following conditions are equivalent:(i) The mapping f is weakly lower semicontinuous?(ii)For every opensets A in ( ) , are semiopensets(iii) For every closed sets B in ( ) ,, are semiclosedsets(iv)For every x ? X ,every ? ? opensets G in ( ) and ,the who arbitrarily converges to x is in .(4) Let (X,?)and(Y,?) be topological spaces , if X→P₀ is a setvalued mapping, then the following conditions are equivalent:(i) The mapping f is weakly lower semicontinuous (ii)For every subset net{A : n D}in Y, there have(iii) For every subset net{A : n D}in Y, there have(5) The mapping X→P₀ is weakly lower semicontinuousif and only if is weakly lower semicontinuousf(6)Let (X ,?) be topological spaces and (Y ,d ) be metric spaces,then the mappingX→P₀ is weakly lower semicontinuousif and only if for every And and ( ) , there exists a semiopenneighborhood of and have(7) Let (X , ?)be topological spaces and (Y ,d ) be normed space, if the mappingf : X P(Y) is weakly lower semicontinuous,then convex mapping is also weakly lower semicontinuous.(8)Let (X,?)and(Y,?) be topological spaces, if X→P₀ is a setvaluedmapping, then the following conditions are equivalent:(i)The mapping f is weakly upper semicontinuous(ii)For every ,there have(iii)For every ,there have(9)Let (X , ?)and(Y , ?) be topological spaces, if X→P₀ is a setvaluedmapping, then the following conditions are equivalent:(i) The mapping f is weakly upper semicontinuous?(ii)For every opensets A in ( ) , are semiopensets(iii)For every closed sets B in ( ) , are semiclosedsets(iv)For every x ,every opensets U in ( ) and ,the net who arbitrarily S converges to x is in f * (U ).(10)Let (X , ?)and(Y , ?) be topological spaces, if X→P₀ is a setvaluedmapping, then the following conditions are equivalent:(i) The mapping f is weakly semicontinuous?(ii) For every ? ? opensets A in ( ) , and are both semiopensets(iii) For every ? ? closedsets B in ( ) and are both semiclosedsetsin(11) LetY be normal spaces, if the mapping X→P₀ is weaklyuppersemicontinuous, then f is also weakly upper semicontinuous.2 (1) Let (X , ?)and(Y , ?) be topological spaces, if X→P₀ is a setvalued mapping, then the following conditions are equivalent: (i) is weakly lower continuousf(ii) For every ( ),there have(iii) For every ? ? closedsets A in ( ) , are closedsets in(iv) For every ? ? opensets B in ( ), are opensets in(2)Let (X , ?)and(Y , ?) be topological spaces, if X→P₀ is a setvaluedmapping, then the mapping f is weakly lower ? ? continuous if and only if for every( ) ,there have .(3)Let (X , ?)and(Y , ?) be topological spaces, if X→P₀ is a setvalued mapping, then the following conditions are equivalent:(i) f is weakly lower continuous?(ii)For every D there have ((iii) For every E ,there have f Eo f E(4)Let (X , ?)and(Y , ?) be topological spaces, if X→P₀ is a setvaluedmapping, then the following conditions are equivalent:(i) f is weakly lower ? ? continuous?(ii) For every subset net{A : n D}in Y, there have(iii)For every subset net{A : n D}in Y, there have(5)Let (X , ?)and(Y , ?) be topological spaces, if X→P₀ is a setvaluedmapping, then the following conditions are equivalent:(i) f is weakly upper continuous?(ii)For every here have(iii) For every B ? ? ,there have...