In Hyperspace Contraction Homomorphism And Continuous Research

In this paper,we have the following conclusions: 1.Finite product of ANR's are ANR. 2.Every contractible space is path-connected. 3.Each partition of unity on X is countable. 4.If X is an AR, then it is also an ANR. 5.Every retraction of AR's is an AR and every neighborhood retraction of ANR's is an ANR. 6.In hyperspace 2X , f :(2~X×2~X ,2~T×2~T )→(2~X ,2~T ) is continuous mapping ,define mapping h *: ? (2~X×2~X ,2~T×2~T )→Π(2~X ,2~T ),then h* is homomorphism. 7.In hyperspace 2X , the finite topology ,if mapping f is continuous ,then we have that f is induced homomorphism of the finite topology. 8.Let 2~X ,2~Y ,2Z be hyperspaces ,02~X×2X,02 y ? 2~Y,02 z ? 2Z, if 0 0? :(2~X ,2~X )→(2~Y ,2 y), 0 0? :(2~Y ,2 y )→(2 Z,2 z) and 0 0h :(2~X ,2~X )→(2 Z,2 z) are all continuous mapping then : (1) 0 01X :(2~X ,2~X )→(2~X,2~X) is induced homomorphism 0 0(1X ) *: ? (2~X ,2~X )→Π(2~X,2~X)is constant homomorphism (2) if h ? ? ? ?, then h* ? ?* ? ?*. 9.Let 1 1(2~X ,2~T ),2 2(2~X ,2~T ) be hyperspaces,1 1 2 2f :(2~X ,2~T )→(2~X ,2~T ).Then the follwings are equivalence : (1) In hyperspaces, C is a random regularly semi-open set of 2X , f *(C ) is a semi-open set of X ; (2) f is LSC; (3) In hyperspaces, D is a random open set of 2X , f * ( D?0 ) is a open set of X ; (4)C is a random set , C ?22~T , f * (C )→( f * (C ?0)). 10.If ( 2X ,2~T )is hyperspace, D? 2X, ? :(2~X ,2~T )→(2~X ,2~T ).In hyperspace , (1) D is a random open set of 2X , (2) ? * ( D?0 ) is a semi-open set of X , (3) ? is LSC. Then (1) and (2) ?(3). 11.If f :(2~X×2~X ,2~- ? 2~- )→(2~X,2~-), D is a random regularly semi-closed set of 2X , f *( D )is a semi-closed set of X , then f is almost LSC. 12.If f :(2~X×2~X ,2~- ? 2~- )→(2~X,2~-), D is a random regularly semi-closed set of 2X , f *( D )is a semi-closed set of X ,then f is LSC. 13.Let (2~X ,2~T ),(2~Y ,2~T )be hyperspaces, N :(2~X ,2~T )→(2~Y ,2~T ) is mapping,then N is continuous mapping ? m is a random open set of (2~Y ,2~T ), N ?1 ( m) is a open set of (2~X ,2~T ). 14.If N :(2~X ,2~T )→(2~Y ,2~T ), M :(2~Y ,2~T )→(2 Z ,2~T ) are continuous mapping ,then M ? N :(2~X ,2~T ) ?(2 Z ,2~T )is also continuous mapping. 15.If N :(2~X ,2~T )→(2~Y ,2~T ), M :(2~Y ,2~T )→(2 Z ,2~T ),02X×2X , N (2~X ) ? 2~Yare continuous;then M ? N :(2~X ,2~T ) ?(2 Z ,2~T ) is also continuous.