PRE-IMAGES OF θ-spaces

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1 Voume 9, 1984 Pages PRE-IMAGES OF θ-spaces by Eise M. Graber Topoogy Proceedigs Web: Mai: Topoogy Proceedigs Departmet of Mathematics & Statistics Aubur Uiversity, Aabama 36849, USA E-mai: ISSN: COPYRIGHT c by Topoogy Proceedigs. A rights reserved.

2 TOPOLOGY PROCEEDINGS Voume PRE-IMAGES OF 8-SPACES Eise M. Graber A mappig from a topoogica space X oto a topoogica space Y is a (quasi-) perfect mappig provided that the mappig is cotiuous, cosed ad the iverse images of poits are (coutaby) compact. The iverse images of topoogica spaces uder both of these types of mappigs have bee of cosiderabe iterest sice the itroductio of such mappigs i topoogy. Eary importat resuts were the characterizatios of (1) perfect pre-images of competey metrizabe spaces as paracompact Cech-compete spaces by Froik [Fr] ad of (2) the perfect pre-images of metrizabe spaces by Arhage'skii [A ] as paracompact 2 p-spaces. These ed to a umber of deveopmets i geeraized metric space theory. We view the resuts we have obtaied as geeraizatios of these which expicate some of the fudametas ivoved. We cosider here the pre-images of a-spaces uder perfect ad quasi-perfect mappigs. I particuar, we defie the cass of pre-a-spaces, which is a more icusive cass tha the cass of D-spaces of Bradeburg [B] ad the *-spaces of Isiwata [I] which characterize the quasiperfect pre-images of T deveopabe spaces. We deote the set of atura umbers by N. Ay referece give beow for a certai cocept cotais a defiitio of the cocept but is ot ecessariy the origia source.

3 52 Graber Defiitio 1. Suppose that (X,T) is a topoogica space ad h: N x X ~ T is a mappig such that for a x E X ad E N, x E h(+,x) ~ h(,x) ad g(,x) = U{h(,z) : z E h(,x) ad x E h(,z)}. If for each x E X, {g(,x): E N} forms a base at x the X is a 8-spaae ([H], [W]). We wi ca h a 8-fuctio provided h ad the associated fuctio 9 are described as above. Coditio 2. Assume that (X,T) is a topoogica space ad that h: N x X ~ T is a mappig such that for x E X ad E N, x E h(+,x) c h(,x). Defie g(,x) = U{h(,z): z E h(,x) ad x E h(,z)}. I Lemmas 2-13, we are assumig that the foowig properties hod. (01) For each x E X, ENg(,x) is cosed. (02) For each x E X, if (y : E N) is a sequece i X such that y E g(,x) for each E N the (y : E N) custers to some poit i ENg(,x). (RI) For each x E X ad E N there exists men such that for a z E g(m,x) there is ken such that g(k,z) c 9 (,x). (RII) For each x E X ad E N, if z E h(,x) ad x E ENg(,y) the there is ken such that g(k,z) ~ h(,y). (Ee) If ENg(,x) ENg(,y) # ~ the ENg(,x) = EN9 (,y) We wi ca a mappig h a pre-8-fuatio provided h ad the associated fuctio 9 satisfy the coditios of 2. First we wi st~te a series of emmas cocerig a space X with a pre-8-fuctio which eabe us to show that such a space is a quasi-perfect pre-image of a T 8-space.

4 TOPOLOGY PROCEEDINGS ~u~ Lemma 3. The coectio {ENg(,x): x E X} is a partitio of X ito coutaby compact cosed subsets. The ~quivaece reatio associated with the partitio i Lemma 3 may be described by x~y if ad oy if ENg(,x) = ENg(,y). For each x E X deote the equivaece cass associated witp x by x. We the have for a x E X, x- = ENg(,x). Lemma 4. For a x E X, {g(,x): E N} is a base at ENg(,x). For each A ~ X, defie INT(A) as {x E X: there is E N such that g(,x) ~ A}. Lemma 5. For each A c X, INT(A) is ope ad INT(A) U{x-: x E INT(A)}. It shoud be oted here that for a E N ad x E X, g(,x) = INT(g(,x)). INT(B). Lemma 6. For each A,B c X, INT(A B) INT(A) Lemma 7. For each A ~ X, INT(A) = INT(INT(A». We wi deote by X- the set of a equivaece casses of X with respect to the reatio "-" Now we wi defie a map f from X oto X- by f(x) = x- for each x E X. Lemma 8. For eadh A c X, INT(A). f-f(int(a».

5 54 Graber Usig Lemmas 8 ad 6 we may make X ito a topoogica space by takig {f(int(a»: A c X} as a base for the ope sets of X- Lemma 9. The fuctio f: X ~ X- is cotiuous. f(int(f-(b»). It foows from Lemma 4, the remark after Lemma 5 ad the defiitio of the topoogy that X- is first coutabe. If x-, y- are distict eemets of X- the ENg(,x) ~ ENg(,y). Thus by (EC) ENg(,x) ENg(,y) = ~. So y ~ ENg(,x) ad x ENg(,y) ad hece X- is T Lemma 11. The fuctio f: X ~ X- is czosed. Remark. The mappig f is a quotiet mappig sice the mappig is cosed ad oto. [E, Coroary ] INTh(,x) = INTh(,y). Proof. Let x- E X- ad x,y E f-(x-). The ENg(,x) = ENg(,y), so x E ENg(,y) ad y E ENg(,x). Let z E INTh(,x). The there is ken such that g(k,z) ~ h(,x). Thus z E h(,x). By, (RII), there is j E N such that g(j,z) ~ h(,y). Thus z E INTh(,y) ad INTh(,x) ~ INTh(,y). By a simiar argumet INTh(,y) c INTh(,x) ad thus equaity hods. Let T be the quotiet topoogy o X- ad defie r: N x X- ~ T' by r(,x-) = f(inth(,x» where x E f-(x-). For each E N, x E INTh(+,x) c INTh(,x) sice

6 TOPOLOGY PROCEEDINGS Voume x E h(+,x) ~ h(,x) ad INTh(,x) = h(,x) by (RII). So x- = f(x) E f(inth(+,x)) ~ f(inth(,x)) for each E N. Now defie s(,x-) = U{r(,z-): z- E r(,x-) ad x- E r(,z-)}. Notice that for each x- E X- ad E N, f(g(,x)) = U{f:q(,z): f(z) E fh(,x) ad f(x) E fh(,z)} U{r (, z-): z- E r (, x-) ad x- ~~ r (, z- )} = s (, x-). Lemma 13. The quotiet space X- is a a-space. Proof. It suffices to show that for each x- E X-, {s(,x-): E N} is a base at x-. Let x- E X- ad et W be a ope set cotaiig x-. Now for each x E f-(x-), -1 x E ENg(,x) =f (W) ad by Lemma 3, {g(,x): E N} is a base at ENg(,x). Thus there is ken such that g(k,x) =f-(w). Hece s(k,x-) = fg(k,x) =ff-(w) = W. So {s(,x-): E N} is a base at x- ad thus X- is a a-space. We have show that if X is a space with a pre-a-fuctio the there is a quasi-perfect mappig f from X oto a T a-space X-. Now et us cosider the reverse impicatio. We first state two emmas which give us a basic compoet of our characterizatio. These are essetiay kow, the eariest referece cocerig cosey reated resuts is [Mo, Chapter V, Theorems 2 ad 3]. Lemma 14. If f: X + Y is a cosed surjectio ad y E y has a coutabe base, the f-({y}) has coutabe character. (i.e. A has coutabe character if there is a coectio of ope sets {U : E N} such that if A C with V ope the there is E N with A ~ U = V.J V

7 56 Graber Lemma. 15. If f: X -+ Y is a quasi-perfect mappig ad Y is T ad {o : E N} is a decreasig oca base at y, the if for a E N, x-- E 0-- = f-(o ), the sequece ( x : EN) has a custer poit i f -1 ({y}).. Theorem 16. Suppose Y is a T a-space ad f is a quasi-perfect mappig from a space X oto Y. exists a pre-a-fuctio for x. The there Proof. Let Y be at a-space ad f be a quasi-perfect mappig from a space X oto Y. Sice Y is a T a-space 1 there exists a a-fuctio p for Y. We ow defie a mappig h: N x X -+ T, where T is the topoogy o X, such that for each x E X ad E N, h(,x) = f-p(,f(x)) ad g(,x) = U{h(,z): z E h(,x) ad x E h(,z)}. Notice that g(,x) U{f-p(,f(z)): f(z) E p(,f(x)) ad f(x) E p(,f(z))} f-q(,f(x)). To see that (01) hods, suppose x E X. The f(x) E Y ad {q(,f(x)): E N} is a base at f(x). Sice Y is a T-space, ENq(,f(x)) = {f(x)} which is cosed. -1 The cotiuity of f impies that f cosed. (ENq(,f(x))) is But f -1 (ENq(,f(x») = ENg(,x) ad thus ENg(,x) is cosed. To see that (02) hods, et x E X ad (Y: E N) be a sequece i X such that Y E g(,x) for each E N. The coectio {fg(,x): E N} is a decreasig oca base at f(x) so by Lemma 15 (Y : E N) has a custer poit i ENg(,x). To show that (RI) hods, et x X ad N. Choose z E g(,x). The f(z) q(,f(x)) ad {q(,f(z)): N} forms a base at f(z). Hece there is ken such that

8 TOPOLOGY PROCEEDINGS Voume q(k,f(z» ~ q(,f(x» ad thus g(k,z) c g(,x). Hece (RI) hods. Next we show that (RII) is satisfied. Let x E X ad E N. Suppose z E h(,x) ad x E ENg(,y). Now z E h(,x) impies f(z) E fh(,x) = p(,f(x» ad p(,x) is a ope set cotaiig f(z). Sice Y is a T e-space there is ken such that q(k,f(z» ~ p(,f(x». Sice x E ENg(,y), f(x) = f(y) ad thus p(,f(x» = p(,f(y». Thus g(k,z) = f-q(k,f(z» c f-p(,f(y» = h(,y). Thus (RII) hods. Fiay we show that (EC) hods. If ENg(,x) ENg(,y) ~ ~ the ENq(,f(x» ENq(,f(y» ~~. But y is a T e-space, so ENq(,f(x» = {f(x)} ad ENq(,f(y» = {f(y)}. Thus f(x) = f(y) ad hece -1-1 ENg(,x) = f f(x) = f f(y) = ENg(,y). Thus (EC) hods ad the proof is competed., I order to summarize what we have just show, et us make the foowig defiitio. Defiitio 17. A space (X,T) havig a pre-e-fuctio wi be caed a pre-a-space. If X is a pre-a-space with the additioa property that for each x E X, ENg(,x) is compact the we wi ca X a strog pre-8-space. We may ow restate Lemma 13 ad Theorem 16 as foows. Theorem 18. A space X is a pre-e-space if ad ozy if there is a quasi-perfect mappig from X oto a T 8-space.

9 58 Graber It is easiy see from the previous defiitio that the fooiwg is aso true. Theorem 19. A space X is a strog pre-e-space if ad ozy if there is a perfect mappig from X oto a T e-space. We ext discuss the cass of we-spaces which is arger tha the cass of e-spaces. Defiitio 20. A space (X,T) is a we-space if ad oy if there is a mappig h: N x X + T such that for a x E X ad E N, x E h(+,x) c h(,x) ad if g(,x) = U{h(,z) : z E h(,x) ad x E h(,z)} the if ( Y : E N) is a sequece i X such that Y E g(,x) for each E N the (Y : E N) custers ( [H], [W] ) We wi aso eed the foowig defiitio. Defiitio 21. A space X is a primitive a-space if ad oy if there is a sequece t = ( t : EN) of we ordered ope covers of X such that if F(x,t ) = F(X,t ) for each E N impies that (x : EN) custers to x [R]. It is cear that if had g are mappigs o a space X as described i Defi~tio 20 which aso satisfies (02) the X is a we-space. Thus a pre-e-space is a we-space ad a we-space which is aso a primitive a-space is a primitive q-space [R]. Hece a pre-e-space which is aso a primitive a-space is a primitive q-space. It is cear that every perfect pre-image of a DO-space [FL] (compact sets have coutabe character) is a DO-space.

10 TOPOLOGY PROCEEDINGS Voume Aso it is ot hard to see that quasi-perfect pre-images of first coutabe spaces are q-spaces i the sese of Michae [M] ad perfect pre-images of first coutabe spaces are spaces of poitwise coutabe type (for each x E X there is a compact set K such that x E K ad K has coutabe character [AI]) Notice that a coditio ike (01) wi aways be true for cotiuous pre-images of T-spaces with a give base property because we are cosiderig iverse images of poits ad poits are cosed i a T-space. A property ike (02) wi aways be true for cosed cotiuous quasi-perfect pre-images because of the coutabe compactess of iverse images of poits, the base property of the image space ad the cosedess of the mappig as Lemma 14 shows. It shoud be oted that the work preseted here coud have bee imited to spaces havig a base of coutabe order but sice the uderyig techiques of proof yied resuts for other importat uiformized versios of first coutabiity we adopt a more geera poit of view by provig the basic resut for a-spaces. I further papers we wi show that by addig the appropriate coditios we are abe to get aaogous resuts for more speciaized spaces. If desired, oe coud simpify the method ad restrict cosideratio to spaces havig a base of coutabe order or spaces havig a primitive base. The theorems ad methods of base of coutabe order theory are fudameta i the work.

11 60 Graber This paper represets a portio of a doctora dissertatio writte uder the directio of Professor H. H. Wicke at Ohio Uiversity, to whom we are greaty idebted. Refereces [AI] A. V. Arhage'skii, Bicompact sets ad the topoogy of spaces, Tras. Mosc. Math. Soc. 13 (1965), [A 2 ], O a cass of spaces cotaiig a metric [B] [E] [FL] [Fo] [H] [I] [M] [Mo] [R] [W] ad a ocay bicompact spaces, Mat. Sb. 67 (1965), (Russia). H. Bradeburg, Huebiduge fur die Kass der etwiakebare topoogica Raume, Ph.D. thesis, R. Egekig, Geera Topoogy, Poish Scietific Pubishers, Warszawz, P. Fetcher ad W. F. Lidgre, 6-spaces, Ge. Top. App. 9 (1978), Z. Froik, O the topoogica product of paracompact spaces, Bu. Acad. Poo. Sci. Sere Math., Astr., Phys. 8 (1960), R. E. Hode, Spaces defied by sequeces of ope covers which guaratee that certai sequeces have custer poits, Duke Math. J. 39 (1972), T. Isiwata, Iverse images of deveopabe spaces, Bu. Tokyo Gakugei Uiv. IV Sere Mat. Sci. 23 (1971), E. Michae, A ote o cosed maps ad compact sets, Israe J. Math. 2 (1964), R. L. Moore, Foudatios of poit set topoogy, AIDer. Math. Soc. Cooq. Pub. 13, Providece, R.I., R. E. Ruth, Compete E-spaces ad 6-spaces, Ph.D. thesis, Ohio Uiversity, November H. H. Wicke, Usig 6-space cocepts i base of coutabe order theory, Topoogy Proc. 3 (1978), Sippery Rock Uiversity Sippery Rock, Pesyvaia 16057