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

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L,et rY(r,r) : i\ru{ro} be the one-point compactification of N. ,] where u- i-s open in X, n,r -lf, and Itt. , (r} comprise an open base for the product topology on Z : X x'\'I(ro). ,et ? be tlie set of all open ultrafilters on Z. ). Since the natural function frotn ,I onto X x {c'r} is a homeomorphism, ,Y is a ciosecl nowhere d-ense sui;set of. Z(I). Ilelrlich [30] gives an example of a llausclorff-closecl space w]rich ,nni:::. , i;l&ai .. ii:t-irlr - l**: r:{ ,ilt**1:; a-t,' IV. The ,:iaelr end s ir r I I tjl i'*ay {ec'..

IT}}A #s5he sett. ,. *&fe s'"€se #nfs ffiw. *l s!. "<€i ffir1 :::: $s;e +€{:i1 w*d. i,fP] i.. 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.

Now let Y be the ilisjoint topological sum of Q and x (definecl in the previous para,graph). o-base for Q, and g was clefinecl in the previous paragraph. This example shows that an open subspace of a countab15'-3*ir" space neecl not be a countably-Baire space. -Baire spaces. A space is saicl to have tlne cou,ntabla cha,itt conili,tion if eveq' clisjoint family of nonempty open subsets of -{ is countable. 17. rf eoery pseu,clo-base for x conta,ins a cortn,table pseud,o-coaer, then X ltas th,e countuble ch,a'in canclition.

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