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

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

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Fmages and inverse images of Baire spaces \lrhen stutlying a basic topological concept it is ahvaS,s of interest to known rvhat types of functions preserve that property. \r'e will now see that Baire spaces are preseryecl by both almost continuous feebly open surjections [21] and feeb]y continuous d-open surjections. T-'et f be a function from a space x into a space y. (o). Also / will be called feebly continuous if for opuo "oury subset Tz of I with/-r(v) +9, there is a nonempty open subset u oi x such tlrat u - f-'(7).

C(X) is a Baire space, then (X',{"} is a Baire spa,ce. r-space X, if either 2x or C{X) i,s a Baire sllace in the strong sen\e, then so is X. ft can also be shorvn that if X is pseuclo-cornplete, then so is Jr. Function space-q are not so well behaved. rvith respect to the property of being a Baire space. enoted by C(X, Y). In tlre case that (Y, d) is a bounclecl metric, Cu(X, Y) witl represent' tire rnetric space (C(X,y),A), where a it th* supremun metric on C(X, Y) inclucecl b-v d. v (resp. the topology of pointwise conlrerg'ence) will be clenoted by Cn(,ll, Y) (resp.

Rt is easy to see that a function is feebly contiuous' if and. only if preserves dense subsets. ense subset is dense. Every almost continuous it iunction is feebl5' continuous. 1- rf f is an almost continu,ou,s feebty open fu,ncti,on frorn a Ba,ire space X o+tlo a space T, thett, Y ,is a Bai,re spnce. Proof. ense open subsets of y, and for each i let Un: int/-l(Zo). Since / is feebly open, each f-r(vt) is dense in x. ense in x. ense in x since x is a Baire space. Again, ; € '*lffi W lr/. r,u' Conor,r,my 4,2.

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