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ISBN-10: 0444108602

ISBN-13: 9780444108609

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The (1) The s p a c e (2) If is X and X E be two q i v e n f o l l o w i n s s t a t e m e n t s are t r u e : E-compact. E and Xo is E-compact of c l o s e d subspace X, is Xo then homeomorphic 9 E-compact. (3) The t o p o l o q i c a l p r o d u c t of a n a r b i t r a r y c o l l e c t i o n E-compact s p a c e s (4) of If El (5) If [Xu is 2 Hausdorff s p a c e , e q u i v a l e n t to E E R ( E 1 ) . spaces, (6) X be 2 Proof. E-compact. : afG) then R(E) i s a non-empty f a m i l y then n be a n (Xa : aEG} is of If is @(El) E-compact E-compact.

Cleari s continuous, then is dense i n i s a n e x t e n s i o n of because t h e c o n s t r u c t i o n y i e l d s s p a c e of t h e r e exists a superspace c l u ( X ) by a homeomor- c l u(X); in particular, X o ( X ) and REX c l u ( X ) d e n o t e t h e c l o s u r e of t h a t i s homeomorphic t o X a* phism ly Let g*, : pEx + E by p a r t ( 1 ) . W e i l l u s t r a t e t h e s i t u a t i o n i n t h e f o l l o w i n g commutative diagram. * PEX *Ea = E U X' Now, let H : pEX -f 9 ilEa * d e n o t e t h e p a r a m e t r i c mapping a s s o - c i a t e d with t h e c l a s s (9, Observe t h a t i f o t h e r words, t h e ath is precisely the nEa H(p) = g ( p ) f o r a l l PEX, IIE, Therefore, : acG).

X i s a neighborhood of {zXjxEA o f t h e compact s e t , . . , Z x :. Then A and cover 23 A B U Zx U . . U Z 2 Xk and and Zx The B C Z'. X h a s a f i n i t e subcover, a r e contained i n the d i s and Zi 9 2; 1 3.. n ZJZ , k 2 This proves ( 3 ) . For any Hausdorff s p a c e t h e property of normality i s X, e q u i v a l e n t t o t h e c o n d i t i o n t h a t any two d i s j o i n t c l o s e d s e t s be c o m p l e t e l y s e p a r a t e d ( s e e Gillman and J e r i s o n , 3 D , page 4 8 ) .

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