Dr. Ivan.Loncar UDK: 51 Znanstveni rad CONTINUITY OF THE TYCHONOFF FUNCTOR

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1 Dr. Ivn.Loncr UDK: 51 Znnstveni rd Fkultet orgnizcije i informtike V r z din CONTINUITY OF THE TYCHONOFF FUNCTOR ABSTRACT. Let C be clss of the inverse systems X = ix A, f~,a~. We sy tht functor F is C-continuous if F(limX) is homeomorphic with lim F(X). In the present pper the continuity of Tychonoff functor. is investigted. Section Two contins some theorems concerning the non-emptyness nd w-compctness of the limit of inverse systems of w-compct spces. Section Three is the min section. Some theorems concering C-continuity of the Tychonoff functor. re proved, where C is clss of the inverse systems of w-compct,.-compct, H-closed or R-clsed spces. Mthemtics 46H40 subjectclssifiction (1980): Primry 54H25, Secondry Key words nd phrses: functor; inverse system, continuity, w-compct,.-compct o. INTRODUCTION 0.1. The set of ll continuous,.rel-vlued (bou~ded) function on topologicl spces X will be denoted by C(X) (C (X)). Unliess otherwise stted, no seprtion xioms will be ssumed A set A ~ X is regulrly closed (open) if A = lnt A (A = lnt A) A set A ~ 2) is sid to be zero-set if there is n f E C (X) such tht A = f (0). The zero-set of f is denoted by Z(f) or by ZX(f). 213

2 Loncr I. Tychonoff Functor L Zbornik rdov (1990), 14 A cozero-set is complement of zero-set. It is well-known [3] tht (i) z(f) = z ( I f I ) = zein) = z ( I f I Ai) (ii) Evry zero-set is Go (iii) z (~g) =2z(f) v z(g) (iv) z (f' + g ) = Z ( I f I + I g I ) = z(f) ()z(g) (v) The countble intersection of zero-set is zero-set Two subsets A nd B of X re sid*to be completely seprtted in X if there exists function fee (X) such tht f'{x ) = 0 for ll x E A, nd f'{x ) = 1 for ll x E B A spce X is sid to be completely regulr [3] provided tht it is Husdorff spce such tht ech closed set F ~ X nd ech x ~ F re completely seprted A spce X is sid to be lmost regulr [9] if for ech regulrly closed F c X nd ech x E X\F there exist disjoint open sets U nd V such tht x E U nd F c V By cf (A) we denote the cofinlity of the well-ordered set A i.e. the smllest ordinl which is cofinl in A We sy tht spce X is qusicompct if every centred fmily of closed subsets of X hs non-empty interesection A spce X is functionlly Husdorff of for ech distinct points x nd y of X there is continuous funct ion f : X ---> [0,1] such tht f(x) = 0 nd fey) = 1. Ech functionlly Husdorff spce is Husdorff It follows tht in functionlly Husdorff spce X for ech distinct points x nd y there re cozero-sets U nd U such x E x Y ~ y ~ nd y E U ~ X - ~ x ~. y 214

3 Loncr I. Tychonoff Functor L Zbornik rdov (1990), If U is cozero-set contining x E X, there exist cozero-set V 3 x such tht x E V ~lv c U. Nmely, if f : X ---> [O,lJ is function such tht x E f ([0,lJ) = U, then we define function F [O,lJ --> [O,lJ such tht F(y) = 0 for y :S f'{x ) j 2 nd F (y) = ((2y - f(x) : (2 - f'i x ) for y > f (x) j 2. Now, let -1 G = Ff. We hve G (0,1) ~ u If X is functionlly Husdorff, then ~x~ = ~ ~U U is the cozero-set contining x E X~. The proof holds from 0.8., 0.9. nd FUNCTOR L Let X be topologicl spce. We define n equivlence reltion p on X such tht x p y iff f(x) = fey) for ech f E C(X). Let L (X) = XjL be set of ll equivlence clsses equiped with the smltest topology in which re continuous ll functions g such tht g. LX E C(X), where LX : X ---> XjL is the nturl projections. In [3:41J is ctully proved tht L (X) is completely regulr. By [x] we denote the e~uivlence clss contining x X LEMMA. If f : X ---> Y is continuous mpping into completely regurl spce Y, then there exist continuous mpping g L(X) ---> Y such tht f = g. LX' Proof. If x J y then must be f'lx) = fey) since f'{x ) fey) implies tht there is f' E C(Y) such tht f'(x) = 0, f'(y) = 1. This is in contrdiction with x p y since ff' E C (X). This mens tht for x' L (X) one cne define g(x') = f(x), x E x' COROLLARY. If f : X ---> Y is cont inuous mpping, then ther exists continuous mpping L (f) L (X) ---> L (Y) such 215

4 Loncr I. Tychonoff Functor. 2bornik rdov (1990), 14 tht. (f).x = Y f LEMMA. If X is functionlly Husdorf, then. X is one-to-one. X --->.(X) Pro 0 r. Trivil. An open set U ~ X is.-open is U is the union of the cozero-sets. We sy tht spce X is w-compct [4] (qusi-h-closed) if for ech centred fmily ~U : E M~ of.-open (open) sets U ~ X the set ~ ~U : E M~ is non-empty THEOREM. If X is w-compct, then. (x) is compct spce ( = T2 qusi-compct). Proof. It suffices to prove tht.(x) is qusi-h-closed since ech regulr H-closed is compct. Let ~U : E M~ be centred fmi ly of open sets in.(x). This mens U is.-open in X. It follows tht ~ ~U : E M~ ~ 0, where U is closure in X. Let x E ~U : E M~. From the continuity of.x we hve.x (x) E ~ ~U E M~ where now U is closure in (X). The proof is completed. A spce X is sid to be.-compct [4] iff ech cover ~U : E M~ of X consisting of the cozero-sets U hs finite subcover THEOREM. If X is.-compct, then. (X) is compct. Proof. Trivil since ech open set in (X) is.-open in X. A spce X is s~9. to be perfectly w-compct (. -compct, H-closed, R-closed) if.x (y) is copmct for ech y E (X) i.e. every equi vlence clss [y] is compct. 216

5 Loncr I. Tychonoff Functo~ L Zbornik rdov (1990), INVERSE SYSTEMS OF W-COMPACT AND L - COMPACT SPACES We strt with the following theorem THEOREM. Let ~ = ~X' f (3' Ar be n inverse system of L -compct (w-compct) funct ionlly Husdorff spces X. If X, E A, re non-empty, then X = lim X is non-empty. Moreower, if f (3 - re onto, then the project ions fx ---> X ' E A, re onto mppings. Proof. From 1.2. it follows tht ~L = ~L (X ), L (f (3), Ar~s n inverse systems. In view of Lemm 1.3. there is mpping L : X ---> X such tht L = (LX -L identity mpping. The mpping X ---> L (X )) nd LX ' E A, is L induces mpping 1im L lim X ---> lim X which is 1-1. This mens tht lim X ::;: 0 iff L O. Since X - is the inverse system of compct spces L hve lim L (~) ::;:O. The proof is completed. Since ech qusi-h-closed spce is w-compct, we hve 2.2. THEOREM. LET X = ~X, f (3' A r be n inverse functionlly Husdorff non-empty qusi-h-closed spces = lim X is non-empty. lim X ::;: -T (X ex) \hie system of X. Then X We sy tht regulr (lmost regulr) spce X is R-closed (AR-closed) if it is closed in ech regulr (lmost regulr) spce in which it cn be embedded [9J. Ech completely regulr R-closed (AR-closed) spce X is compct since X c (3 X [2J. If X is R-closed, Y regulr, nd f X ---> Y continuous mpping then Y is R-closed THEOREM. Let X JX -1 ' f(3' A r be n inverse system of 217

6 Loncr I. Tychonoff Functor ~ Zbornik rdov (1990), 14 non-empty functionlly Husdorff R-closed spces X. Then X = lim X is non-empty. P roo f. The spce ~ (X ) is completely regulr R-closed i.e. Husdorf compct spce. See the proof of Theorem We sy tht mpping f : X ---> Y is ~ -open if feu) is ~ -open for ech ~ -open set U c X THEOREM. Let X = ~X, f (3' A~ be n inverse system of w-compct functionlly Husdorff spces X If the projections f : lim X ---> X, E A, re ~ -open, then X = lim X is functionlly -, - Husdorff nd w-compct. Proof. Let U = ~U : M E M~ be mximl centred fmily of ~ M -open sets in X. For ech E A let U = ~f (U) : M E M~. We M prove tht U is the mximl centred fmily of ~ -open sets in X (f is ~ -open!). Suppose tht V is ~ -open in X such tht V n -1 f (U) is non-empty for ech U E U.This mens tht f (V) is M M ~ -open set wich meets ech U. From the mximl ity of U it -1 M follows tht f (U) E U i.e. V E U. Hence, U is mximl. From the w-compctness of X it follows tht Y = n ~r-ro-) : U E M M is non-empty. From the mximlity of U it follows tht U contins ll neighborhoods of ll y E Y From it follows tht Y = ~y ~, where y EX. For ech E A let W be fmily of ll ~ -open sets contining Y. From the mximlity of U(3,(3~ it follows tht U(3contins f:~ ( U) = ~f:~ (U) : U E U~ This mens tht f(3 (Yf3) = Y' (3~. Hence y = Y: E A~ is point of X. It is redily seen tht yen ~U : U E ~. The proof is completed since it is cler tht X is functionlly Husdorff. U~ 218

7 Loncr I. Tychonoff Functor ~ Zbornik rdov (1990), THEOREM. Let X = ~X, - IX perfect w-compct (~-compct, spce X = 1im X is non-empty non-empty. 3. CONTINUITY OF THE FUNCTOR ~ f IX (3'A~ be n inverse system of H-closed, R-closed) spces X. A IX iff the spces X, IX E A, re IX Let ~ = ~XIX'fIX(3'A ~ be n inverse system nd let ~ be Tychonoff functor described in Section One. From it follows tht ~(X) = ~~(XIX)' ~(fix(3)'a~ is n inverse system. Let C be clss of the inverse systems. We sy tht the functor ~ is C - c o n tin: u 0 u s if ~(l im X) is homeomorphic to 1im ~(~) for ech X in C. The functor ~ is sid to be continuous if ~ is C - continuous for ech clss C 3.1. LEMMA. If X is n inverse system, then there exists continuous mpping ~1 : ~ (lim X) ---) lim ~ (X). Proof. Let ~ = ~XIX'f IX (3'A~ be n inverse system nd let ~ (X = ~~(X&, ~(fixs' A~. From 1.2. it follows tht there is ~1 'r (lim X) ---) - IX (X ) such tht ~ f = ~1 ~, where ~ lim X ---) ~ IX X IX IX IX (Li m ~). It is redily seen tht ~1 = ~(f(3). ~1 ' (3 ~ IX. This IX (3 mens tht the mppings ~1 ' IXE A, induce continuous mpping ~1 IX : ~(lim X) ---) lim ~(X). The proof is completed LEMMA. lim ~ = ~1 ~ Proof. From the definition of ~1 it follows ~1 = f~ ~1' where IX 219

8 Loncr I. Tychonoff Functor T Zbornik rdov (1990), 14 f' : lim T (X) ---) T(X ) is projection. Moreower, TX f = T1 T - nd TX f = f~. lim T. It follows tht T1 T = f~ lim Lnd Ll T = f~. T1. T i.e. lim T = T1T. Q.E.D THEOREM. Let C be the clss of ll inverse systems X = ~X, - f (3' AL such tht X, E ( A, X = lim X- is w-compct (T-compct) functionlly Husdorf. If the projections f : X ---) X, E A, re onto, then the Tychonoff functor T is C -cointinuous. Proof. From Lemm 1.3. it follows tht ech TX ' E A, is 1-1. This mens tht Iim T is 1-1. Since Iim ~ is funct ionlly Husdorf we infer by 1.3. tht T : lim X ---) T (lim X) is 1-1. It follows tht T1 : T (lim X) ---Y lim T (X) is one-to-one. Since lim T (~) nd T (lim ~) re compct (1.4.THEOREM) we infer tht T 1 is homeomorphism. The proof is completed COROLLARY. Let C be the clss of ll inverse systems n in Theorem 2.4. Then the Tychonoff functor T is C -continuous REMARK. In [4J is proved tht if ~X : E A~ is fmily of w-compct spces X ' then n X is w-compct n T (nx ) = n T (X ) THEOREM:,...«H-closed (R-closed). If the Let H be clss of the inverse systems X = JX, - 1 f Q A~ such tht X E A, X = lim X re functionlly Husdorff projections f : X ---) X E A, re onto mppings, then the functor T is H -continuous. Proof. The spces T(X ), E A, nd the spces T (lim ~), lim ex T (X) re compct (See the proof of 2.3. nd 3.3.).

9 Loncr I. Tychonoff Functor L Zbornik rdov (1990), 14 In [14] it is proved tht lim ~ is H-closed if X re H-closed, f{3 open nd tht f{3 re onto if f{3 re open onto. Hence, from 3.6. we obtin THEOREM. Let H be clss of the inverse system of H-closed functionlly Husdorff spces X nd open onto mppings f Q Then,J the functor L is H -continuous. From [6] it follows tht lim X is R-closed (AR-closed) f X re R-closed (AR-closed) nd if f {3 re open-closed. By similr method of proof we hve THEOREM. Let R-closed (AR-closed) open-closed onto mppings R be clss of the inverse systems of functionlly Husdorf spces X nd Then the functor L is R -continuous. We sy tht n inverse system ~ = ~X ' f{3' A r is fctorisble (or f-system) [10] it for ech continuous mpping f : 1im X---> [0,1] there exists continuous mpping g : X ---> [0,1] such thf f = g f' where,f: 1im X ---> X is the nturl projection LEMMA. If ~ is n f-system, then the mpping L1 : L (lim X) ---> lim L (X) is one-to-one. Proof. Let [x] nd [y] be two distinct points of L (ltm ~), where x, y E lim X. This mens tht there exists n f lim X ---> [0,1] such tht f (x) = nd f (y) = 1. Since ~ is f-system there is n E A nd g : X ---> ro,l] such tht f = g f. It L follows tht [f (x)] *- [f (y)] since g f (x) = nd g f (y)=1. This men~ tht L 1([x]) *- L1([y]. The proof is completed. 221

10 Loncr I. Tychonoff Functor < Zbornik rdov (1990), THEOREM. Let W be clss of the inverse f-system X = ~X' f Q' p A l such r tht ll X nd X = 1im X re w-compct - (H-closed, <-compct, R-closed, AR-closed). Then the Tychonoff functor < is W -cotinuous. Proof: From 1.4. Theorem it follows tht < (lim X) nd lim < (~) re compct. By virtue of 3.5. Lemm it follows tht <1 is the homemorphism Q.E.D LEMMA. [llj. Let ~ = ~X' f [3' A~ be well-ordered inverse system such tht w (X ) < < nd cf'{a) > < > ~ If f Q re 0,-,

11 Loncr I. Tychonoff Functor. Zbornik rdov (1990), 14 Eerfect (open or X is continuous) then w (limx) <. We close this Section with the following THEOREM. Let C be clss of the inverse systems_ X s in If lim ~ is w-compct (.-compct, H-closed, R-closed, AR-closed) dn if the projections f : X ---> 0: E A, re onto, 0: then the functor. is C -continuous. Proof. In view of Theorem it suffices to prove tht X is n f-system. Let X = lim X nd let f : X ---> [0,1] be rel-vlued function. For ech Z E [0,1] let N z be countble fmily of open sets such tht n ~U : U E Nz~ = ~z~. We cn sume tht N = ~N : z e [0,1]~ is countble. It is redily seen tht z -1 for ech U. E f (N) there exist n 0: E A nd open U ~ X such I 0:. 0:. -1 I I tht U. = f (U) [7] (See lso [12]). Since the crdinlity I 0:. 0:. I I INI~ ~ nd cf (A) > N there exist n 0: E A such tht 0: > 0:., i E o 0 I N. Let Y be set n JU : f- 1 (U ) E f- 1 (N ) L. It is cler tht Y z 1 0: 0: 0: 0: Z ( Z n Y ' = 0 z iff z:;t z' nd tht Xo:= u ~Yz : z E [0,1] ~.This mens tht for ech x E X there is only one z E [0,1] such tht x E 0: 0: 0: Y. Put g (x ) = z. We define g:x ---> [0,1] such tht f = g f. Z 0:/0:: 0: 0:: 0::0: In order to complete the proof we prove tht g is continuous. Let 0:: x E X nd let g (x ) = z. For ech neighborhoods V e N there is 0: 0: 0: 0: -1 z neighborhood U of x such tht f (U) = V. This mens tht g 0: 0: 0: 0: 0: (U ) = V. The proof is completed. 0: 4. CONNECTEDNESS OF THE LIMIT SPACE We strt with following theorem 4.1. THEOREM: A topologicl spce X is connected iff.(x) is connected.

12 Loncr I. Tychonoff Functor. Zbornik rdov (1990), 14 Proof. If X is connected, then. (X) is connected since.x : X ---) (X) is continuous surjection. Conversely, let.(x) be connected. If X is disconnected, then there exist two disjoint open sets V, V~ X such tht X = V u V. Let g : X ---) [O,lJ be mpping such tht g (x) = if x E V nd g (x) = 1 if x E V. Clerly, g is continuous. From the definition of (X) it follows tht.x(v) n LX (V) = 0 nd.x (V) U.x (V) = L (X), where LX (V) is the imge of V. Let f : L (X) ---) [O,lJ be mpping such tht f[lx (V)J = 0, f[.x (V)J 0= 1. Clerly, f =g. Since g E C (X), from the definition fo (X) it follows t~rt is continuous i.e. f E C (. X». This mens tht L (V) = f (0) x nd. (V) = f -1(1) i.e.. (V) nd L (V) re disjoint open sets x x x in (X). This contrdiction with the connectedness of (X). The proof is completed THEOREM. Let ~ = ~X' f~' A~ be n inverse system such tht the functor is_x-conti~uous. The spce X = lim X is connected iff lim X is connected. Proof. The spce L (1im X) is connected since it is homemorphic with 1im X. From 4.2. it follows tht 1im X is connected iff. (lim Xl-is connected. Q.E.D. Now, from Theorems 4.1. nd 4.2. nd from theorems of Section Three we obtin the following theorems THEOREM. Let X be n inverse system s in Theorem 2.4. Then X = lim X is connected iff X E A, re connected. Ar be n inverse system such tht 4.4. THEOREM. Let X = JX, f Q' - 1,... X, E A, X = lim X re (R-closed). functionl~y Husdorff H-closed If the pr-o.ject rons f : X ---) X E A. ', re onto

13 Loncr I. Tychonoff Functor L Zbornik rdov (1990), 14 mppings, then X is connected iff X, E A, re connected THEOREM. Let X = JX, - ") f(3' A ~ be n inverse system of H-closed funct ionlly Husdorf spces X nd open onto mppings f(3' The spce X = 1im X is connected iff X, E A, re connected THEOREM. Let X = ~X' f (3' A~ be n inverse system of R-closed (AR-closed) functionlly Husdorf spces X nd.open-closed onto mppings f(3' The spce X = 1 im X is connected iff the spces X, E A, re connected THEOREM. Let ~ = ~X' f (3' A~ be n inverse f-system such tht ll X nd X = 1im ~ re w-compct (L-compct, H-closed, R-closed, AR-closed). X is connected iff X, E A, re connected. REF ERE N C E S 1) Arhngel'skij A.V., Ponomrev V.I., Osnovy ob{~ej topologii v zd~h i upr'nenijh, Nuk, Moskv, ) Engelking R., Generl Topology, PWN, Wrs zw, ) Gillmn L. nd Jerison M., Rings of Continuous Functions, D. vn Nostrnd Comp., New York, ) Ishi it., Some result s on w-compct spces, (in russ in), UMN 35 (1980), e 5) Lon+r- I., Applictions of e-clsed nd u-c losed sets, Zbornik rdov Fkultet orgnizcije i informtike Vr'din 8 (1984), (6), Inverse systems of R-closed spces, Mth. Blkn., 2(1988), (7), LindelOfov broj i inverzni sistemi, Zbornik rdov FOI Vr.'din, 7 (1983), ) Sigl M. K. nd Ash Mthur, A note on lmost complete ly 225

14 Loncr I. Tychonoff Functor L Zbornik rdov (1990), 14 regulr spces, Glsnik mtemti~ki 6 (26) (1971), ( 9), On miniml lmost regulr spces, Glsnik mtemti~ki 6 (26) (1971), (10) [~epin E.V.:, Funktory i nes~etnye stepeni kompktov, UMN 36 (1981), (11) Tk~enko M.G., Some results on inverse spectr I., Commenttiones mth. univ. crol. 22 (1981), (12), Some results on inverse spectr II., Commenttiones mth. univ. crol. 22 (1981), (13), Cepi i krdinly, DAN SSSR 239 (1978), (14) Vinson T.O. nd Dickmn R.H., Inverse limits nd bsolutes of H-closed spces, Proc. Amer. Mth. Soc. 66 (1977), Primljeno: Loncr I. Neprekidnost Tihnovljevog funktor SAD v R Z A J U rdu je istrzivn neprekidnost Tihonovljevog funktor L. Pri tome kzemo d je funktor F C-neprekidn ko su prostori F(lim X) i limf ~ homeomorfni, gdje je C kls inverznih sistem X - - ~X,f{3'A~. U odjeljku 1. dn je definicij i osnovn svojstv funktor L. Drugi odjeljk sdrzi teoreme 0 neprznosti i w-kompktnosti limes inverznih sistem w-kompktnih prostor. Teoremi iz drugog odjeljk sluze z dokzivnje teorem 0 C-neprekidnosti funktor L, gdje je C kls inverznih sistem w-kompktnih (L-kompktnih, H-ztvorenih ili R-ztvorenih) prostor. 226.