Baire spaces by R. C. McCoy, R. A., Haworth

By R. C. McCoy, R. A., Haworth

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19. couerforX. pa'i'rwi,se d'i,sjoitt't (111) Eaerg ltseud,o-base for x conta'ins a cou,ntable pseud'o-couer for X. 20. x ,is a countublg-Bai,re spaca i,J and, only i'f at I'oust one of the fot'lorni'ng two proporti,es hol'd'z (i) X is a Baire spu'ce. efinition of countably-Baire space is vacuously The next propositions follow from Theorem 3'20' PnoposrrroN 3'21. roof' spaces 'is a cou,ntabl'g-Bai're sPacl' i,s a cogrztablg-Baire sgtuce wlt'iclt is tt'ot a Ba'ire i,s a coun'tably-Ba'ir'e space.

Rhen, there er'ists a d,ense subspace D of X such that (i) each f*lo is conti,nttous, and (ii) for each open subset U of Wn, g"(U) ,is optett, 'i,n D. Blumberg type theorems may arise while stud,ying special types of functions. This has been the case for c-continuous functions [22], semi-continuous functions ([16], pp. 111, 1-1-4), ancl functions for which the inverse image of open sets are Jf" ([16], p. 110). L,evy l38l has introduced the notion of strongly non-Biumberg spaces and- investigated.

Spaces have a pseudo-base that is the countable union of disjoint families. 1] gives a Blurnberg type theorem for such spaces. 5. Let x haae a o-d,i,sjoint psaudo-base, and,l,et T be a second, countable space wh'ich conta,ins an i,nfi,nite d,i,suete subsat. rhen x is n, Baire space i,f and, onlg if i,thas Blu,mberg,s property w,ith respectto T. Proof. I-tet g : () ,1. $ {*,,1 m<-l[] be a pseudo-base for -ltl, where aach 9n is a disjoint famiiy, and. let Mo : g. ucih that' 9r - flr. Suppose that for each n,: 1r .

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