Open and Proper Maps Between Convergence Spaces

8/2/2019 Open and Proper Maps Between Convergence Spaces

1/10

Czechoslovak Mathematical Journal

Darrell C. Kent; Gary D. Richardson

Open and proper maps between convergence spaces

Czechoslovak Mathematical Journal, Vol. 23 (1973), No. 1, 15--23

Persistent URL: http://dml.cz/dmlcz/101140

Terms of use:

Institute of Mathematics AS CR, 1973

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to

digitized documents strictly for personal use. Each copy of any part of this document must contain

these Terms of use.

This paper has been digitized, optimized for electronic delivery and stamped

with digital signature within the project DML-CZ: The Czech Digital Mathematics

Libraryhttp://project.dml.cz
http://project.dml.cz/http://dml.cz/dmlcz/101140


2/10

Czechoslovak Mathematical Journal, 23 (98) 1973, Praha

O P E N A N D P R O P E R M A P S B E T W E E N C O N V E R G E N C E S PA C E SD A R E L L C. K E N T , Pullman and G A R Y D. RICHARDSON, Greenville

(Received September 6, 1971)

I N T R O D U C T I O NThis paper is to some extent a cont inuat ion of [ 2 ] , and we therefore ask the

reader to refer to this source for the notation and terminology not specified herein.In our study of open and proper maps between convergence spaces, we are interestedprimarily in results pertaining to the theory of convergence spaces rather than ingenerahzations of well-known topological theorems. We show, for example, thatopen maps extend from a convergence space to its decomposit ion series whereasproper maps do not; furthermore, open maps preserve the property of pretopologicalcoherence ofpairs of spaces. Alternate characterizations are given for both types ofmaps. In the last section we study the behavior of the extension map from a convergence space to i ts Stone-Cech compactif ication.

1 . P R E L I M I N A R I E SAll spaces considered in this paper areassumed to be convergence spaces. We

denote by C(S) the set of all convergence structures on a se t S.By a mapwe shall mean a continuous onto function. Let / : (S, q) -^ {T , p) be

a m a p ; / is a convergence quotient map {biquotient map) if, whenever a filter (u.f.(ultrafilter)) #" p-converges to j , there is xmf~^{y) and J> mapping on ^ such that J^-converges to x.

For the map / abo ve, let jR be the ass ociated equivalence re lation {{x, y) : f{x) = f{y)}- Let be the canonical m ap of S onto SJR defined for all x in S by 9{x) = [ x ] ,the K-equivalence class containing x. Let qJR be the finest convergenc e struc tureon SJR which makes 9 cont inuous. F inal ly , le t /^ : 5/J^ -T b e defined / ( ) ==m.

Proposition 1.1. {SJR, qJR) isHausdorff if and only if R is closed in the productspace.

15


3/10

Proof. We prove only the sufficency, since this is not vaHd in the topological case.Let #* and J be filters on S\R which ^/R-converge to the dist inct e lements [x], \y \respectively. Since 0 is a convergence quotient map, there is ^^ which maps on J^and ^-converges to x^ in [x] and J^i which maps on J and g-converges to y^ in \y \Since (x j , j i ) ^ R and R is closed, there is F^ in ^^ and G^ in e/^ such that(Fl X G^r\R=-^. Thus (Fi) n (Gi) = 0, and the filters J^ and J are disjoint.Proposition 1.2. The following statements a re equivalent:(a) / 15 a convergence quotient map.(b) /R /S a convergence quotient map.(c) /i^ S a homeomorphism.If (5 , ^) and (r , p) are pseudo-topological spaces and qJ R the finest pseudo-topology

relative to which 0 is continuous, then Proposition 1.1 remains valid if "convergencequot ient map" i s rep laced by "b iquot ient map" .Let {(S^, q^} be a family of convergence spaces, and let S be the disjoint unionof the 5^'s. The disjoint sum (5 , q) of the given family is defined as follows: x G q{^)means tha t ^ contains one of the sets S^, x is in S^ , and the restriction of #" to S^^^-converges to x; we write (5, q) = YJ{S^, q^).a

Proposition 1.3. Let q e C(5) , [q^ a C{S); let ( , p) = ^ ( S , q^). Then the canon-aical map h of ( , p) onto (S, q) is a conve rgence quotient map iffq= in f [q^].

Given a convergence space (S, q) , there is a finest pretopology n(q) on 5 coarserthan q and a finest pseudo-topology a(q) coarser than q; the former is called thepretopological modification and the latter the pseudo-topological modification of q.If p and q are both in C{S), then n{q) = 7i{p) means tha t p and q have the same neighborhood filters, while a{q) = ( ( ) means tha t p and q coincide on u.f. 's.If X is any p oin t in a set S and J^ any filter on S, then we denote by = (x, J^)the finest topology on S relative to which #" converges to x. The neighborhood filtersfor this topology are: v^(x) = #" n x; v^{y) = y for y x .

Proposition 1.4. Let f : (


4/10

2. O P E N M A P SDefinition 2.1. An onto fun ct io n/ : (5 , q) -^ ( , p) is open if it satisfies the following

con ditio n: (A) whenever a u.f. # " on T j7-converges to y, then for each x i n / " ^ ( j )there is a filter ^^ which ma ps on #" and ^-converges to x.Continuous open functions will be called open maps.Proposition 2.2. If f : (S, q) -^ ( , p) is an open function and(S, q) a pretopologica l

space, then f satisfies: (A' ) whenever a filter J> on Tp-converges to y, then for each xinf~^(y) there is a filter J which maps on ^ and q-converges to x.

Proof. Let ^ ^-converge to x. For a given in f~^{x) and for each u. f. J^ finerthan J^, a u.f. maps on and ^ -converges to y; the intersection of these 'maps on ^ and (since ^ is a pretopology) ^-converges to y.

In part icular, every open map with a pretopological domain is a convergencequotient ma p. The next proposit ion, whe n compa red w ith Proposit ion 1. ., exhibitsa certain duali ty between open functions and convergence quotient maps.

Proposition 2.3. Let {q^} cz C(S); le t ( , p) = ^ ( S , q^). If q = sup q^, then thea

canonical function from (T , p) onto (S, q) is open. C onverse ly, if the canonicalfunction is an open map and each q^ is a pseudo-topology, then q = q^^for each a.

Proposition 2.4. / / / : {S, q) -> (T, p) is an open function, then, for each x in S,Vp(f{x)) ^ f{^q{ ^))- If ^he spaces involve d are topological, then f carries op en setsinto open sets.

Proof. Let V ^f{^q{^)) and let #" be a u.f. which p-converges to f{x). Then thereis a u.f. J' which ^-converges to x and maps on #". Since v = f{u) for some e vj^x)and UEJ, vEf{J) = ^ . T h u s v is in each u.f. which p^converges to f(x), a ndv^vXf(x)).

Proposition 2.5. An open map is neighborhood-preserving. A neighborhood-preserving map with a pretopological domain is open.

Proof. I f / : (S, q) -> ( , p) is an open map, then f{vq{x)) = Vp{f{x)) for all xin S by Proposi t ion 2 .4 and the cont inuity of / , and s o / i s neighborhood-preserv ing .Ne xt, assume that / is neighbo rhood-pre serving an d (S, q) a pretopological space;let #" be a u.f. which p-converges to f(x). Then ^ ^ f{^q{^)) and so some u.f. finer than Vq{x) maps on #". Since ^ is a pretopology, / must ^-converge to x.

The next result is an immediate consequence of Proposit ion 2.5 and Theorem 3,[ 2 ] . Following the notation of [2] , we denote by {n^q) : 0 ^ a ^ J the decomposition series for (S, q) .

17


5/10

Corollary 2.6. Letf : (S, q) -> (T, p) be an open map. Then:(1 ) ^ (2 ) If {s, q) is a pretopolog ical (topological) space , then (T, p) is a pretopolog ical(topological) space.(3 ) In the following diagram, where the horizontal arrows are identity maps and

the vertical arrows represent f, all vertical maps are open.(S , q) - . (S , n{q)) -^ .i i( , )^{ , { ))^..

..^{S,n\q))^.I. . - . (ZnXp))-^. ...^{S,X{q)) i, . - ( T , A ( p ) ) .

Let fi : (Si, q^ -> (T^, Pi), i in some index set / , be a family of maps. Let (S, q)a nd (T, P) be the product spaces of the (Si, q^^s and (Ti, Pi)'s, respectively, and let/ : (S, q) -^ (T, P) be defined as follows: f(x) = iff/^( ^) = J i for all i e I, wh ere x^is the p ro ject ion of x onto 5^. T h e n / i s cal led the produc t of the /^ ' s . Op en map s areproductive i n the sense t h a t / i s o pen whenever each of t h e / / s i s o p en ; i ndeed , t hesame can be said for convergence quotient maps and biquotient maps.

A p a i r ( S i , qi) and (52, qi) of convergence spaces is said to be pretopologicallycoherent (see [3]) if the p rodu ct of the pretop ological modifications coincides with thepretopological modif icat ion of the product space.

Proposition 2.7. Let (S^, qi) and (S2, qz) be a pretopologically coherent pair ofspaces , and let (T^, pi) and (T2 , P2) be their images under the open mapsf^ andfz-Then (TI, PI) and (T2 , pz) also form a pretopologically coherent pair.

Proof. L et (S, q) = (S q,) x (^2, ^2), (Z p) = ( ^ Pi) x (T^, P2), (S, r) == (S 7i(q,)) X (^2, n(q2)), (T, s) = (T n(p,)) x (T 2,71(^2)). By Co rollary 2.6,fi : (Si, n(qi)) -^ (Ti, n(pi)) i s an open map for i = 1, 2. T hus / : (S, r) -> ( , s) i san open map by the aforement ioned product iv i ty of open maps . But n(q) = byassumption, so i t fol lows that 7i(p) = s.

Proposition 2.8. Letf : (S, q) -^ ( , p). Thenf is an open map if and only if thereis a set {p^} of convergence structures on T such that (7(p) = cr(inf {p^}) and a maph : ( , p^) -> (SJR, r) such thatfj^ h is the canonical map, where r is the pseudo-topology on SJR defined as follows: a u.f. J' r-converges to [ x ] if for each x^ e [x ] ,there is a u.f. H mapping on which q-converges to x^.

Proof. A ssume the given conditi on, a nd let J^ be a u.f. .which p-con verges to y.T he n # " jp^-converges to for some a. T hus h(^) r-converges to h(y), and since h(^)is a u.f. on SJR, there is for a given x in f'~^(y) a u.f. which ^-converges to x sucht h a t ( ) = h(^). T h u s / ( = ^ ) = # ", a n d / i s an o p en m a p .

For the converse argument one can use the same construction as in the proof ofProposit ion 1.4.18


6/10

3 . P R O P E R M A P S

From the discussion of proper maps in Section 10, [1], we select a characterizationwhich is especially suitable for convergence spaces.Definition 3.1. A m ap / : (S, q) -> (T, p) is proper if f{q{^)) = p{f{^)) for all

u.f. 's ^ on S,In other w or ds , / is p roper if, w henever a u.f . J^ p-converges to y, each u.f. J^ which

maps on # ^-converges to some point x in f~^{y). Thus proper maps somewhatresemble open maps, but with the roles of points and u.f . ' s interchanged. Note thatfor one-to-one maps, the notions proper map, open map, and biquotient map areall equivalent.

Given a convergence space (S, q) and a S, le t ^( ) = A and let F'J^) be theclosure of A. If a is an ordinal number and a 1 exists, let 1( ) be the closureof ri~\A); if a is a hmit ordina l , le t ^ ( ) be the union of ^ ( ) for < a. Note tha tthe least ordinal a such that 1{ ) = ^ ' ^( ) for a l l cz S is what we have beencalling jq , the length of the decomposition series for (S, q) . A set is closed if it coin cideswith its own closure.

Proposition 3.2. Let f : (S, q) -^ (T , p) be a proper map, A cz S, a an ordinalnumber. Then /(F'^q^A)) = ^ ( / ( ) ) . Proper maps preserve closed sets.

Proof. Transfinite induction, a = 1. It is always true for continuous maps that/ ( r^ (A) ) c : [/( )). L et e {/{ )); then there is a u.f. ^ p-converging to whichcontains f{A). N ow A nf'~^{^) has a u.f. refinement ^ , an d J^ ^-converges to xi n / - ^ ( 3 ; ) . Thus X is in , ( ) , a n d / ( , ( ) ) .

Next assume the statement true for all Q < a. If a 1 exi st s, t hen / ( ( ^ ) ) == f i r , { r i ~ \ m = ' \ ) ) ) = ^ { ~ \ ) ) = m m ) - tr is a umitordinal , then /{ )) = f{[j{n{F) : g < a = U r ^ ( / ( f ) ) = {/{ )). ,

The last assertion follows by taking large enough so that r^(F) = T^^^().Given a filter #" on (S, ^), let ^(# ' ) = #". If a is a Hmit ordinal and a 1 exists,

let r^ (# ' ) be the fil ter generated by the collection of closures of mem bers of ^~^ (.^);if a is a Hmit ordinal, let T\{^) = \{ {{^) : < (x).

Proposition 3.3. The following are consequences off being a proper map.(1) For each filter ^ on S and each ordinal number G , fiF^^^)) = ^ ( /( # ') ) .(2 ) (3) / / q is topology, then n(p) is a topology.Proof. (I) Follows from Pro pos ition 3.2 by a simple transfinite indu ction

argument .(2) If Fl'-^B) = Fl{B) for al l S, then i t follows from Prop osit ion 3.2 that

;+ ^( ) = Fl{A) for all A T Thus (2) follows.19


7/10

(3) If ^ is a topology, then 7^ ^ 1 which imphes ^ 1. But y^ ^ 1 if and onlyif n{p) is a topology.By comparing Proposition 3.3 with Corollary 2.6, we see that proper maps havesome properties in common with open maps. However the following example showsthat the image of a pretopological space under a proper map need not be preto-pological; from this it follows that a proper map need not be proper relative to thedecomposition series^of the domain and range spaces.Example 3.4. Let T be an infinite set, x e T, and ^ a free u.f. on T. Let p be thepseudo-topology on T whose convergence is described on u.f.'s as follows: p-con-verges to y, all \ each free u.f. other than #" p-converges only to x; ^ fails toj7-converge. With each free u.f. J^ on distinct from #", associate an element a^

not in Tand let A = [a^] be the set of all such elements. Let 5 = T u , and let qbe the pretopology on S whose neighborhood filters are defined as follows: vj^y) = yfor y , x; vj^x) = x n { a S : :=> A}; vj^a^) = ^ r\ J' for a^ e A,where J' is the filter on S generated by the filter J on T. Finally, let / : (S , q) ->-> ( T , P ) be defined by: f{y) = yor y \ f{y) = x for 3; e . It is easy to see th a t/is a proper map. Note that #'7r(p)-converges to x, but no filter which maps on ^q -converges (i.e., 7r(^)-converges) to a point in /~ ^ (x ), so th a t / : (5 , K{q)) -> (T, n{p))is not even a biquotient ma p.In the preceding example, let be any free non-u.f. which does not have ^ as

a refinement. Then p-converges to x, but no filter which maps on can ^-converge to preimage of x, and it follows that the map / described above is nota converge quotient map, even though the domain space is pretopological.We omit the straightforward proofs of the next two results.Proposition 3.5. / : (S, q) -> (T, p) is biquotient if and only if / ^ : {SJR, qJR) ->-> (T, p) is proper.Let / : (5, q) -> (T, p), and let be the pseudo-topology on SJR defined by speci

fying u.f. convergence as follows: a u.f. J^ ^-converges to [x] if either [x] or,for each u.f. on S such tha t { ) = J, ^-converges to some Xi in [x].

Proposition 3.6. A map f : (S, q) -> (T, p) is proper if and only if there is a set[p^] cz C(T) such that ( ( ) = ^(inf {p^}) and a map h : ( , p j onto {SJR, )such thatfg^ h is the canonical map onto ( , p). *

It can be shown by a direct argument that proper maps are productive. Indeed thegood behavior of proper maps relative to products was used to define proper mapsin [1], and in the next proposition we generahze this characterization to convergencespaces.20


8/10

Proposition 3.7. Let f : {S, q) -^ (T , p). Then f is proper if and only if, for eachconvergence space (Z, r) and each A c: S x Z, {f x i^) { % )) = ^ ( (/ x i^) (A))for all ordinal numbers (x, where s = q x r and = p x r are the product convergence structures.

Proof. If / is pro pe r, the n the given con dition is estabHshed w ith the help ofProposit ion 3.2 and productivity of proper maps. Conversely, let #" be a u.f . on Ssuch tha t / (# ' ) p -converges to y. Let Z = S {w}, where w is some p oint not in S;le t r be the topology (w, J^') (see Proposition 1.4) where ^' is the filter on Z gene ratedby #". Let A = {{x, x):xeS};by hypothesis ( / x i,) { ,{ ) ) = , ({ (/ ( ) , x)\xee S}). kho f{^) ]7-converg es to >% ^ r-converges to w, and [/ (F ) x F ) n ( / x i^ .. ( ) 0 for a l l F e #" , imply ing ( j , w) e ^( ( / x i^) (^) ) . Thus there i s x in S' suchthat (x, w) G 5 ( ) wi th x in / " ^ ( j ) , and there a re filte rs j ^ ^ and J^2 on S such that J> ^^-converges to x and ^2 (rega rded as a filter on Z) r-conve rges to w, with (G^ x G2) nn 0 for all Gl e J^ an d G2 e J^2- Th us J^i v J2 exists and ^-converges to x.But . /2 ^ ^ 5 w hich implies . / 1 v J^2 = -^ J r i^ so #" ^-converges to x, as desired.

We conc lude this section by fisting, without proofs, a few propo sit ions abo ut prop ermaps from [l] which extend without difficulty to the convergence space sett ing. Wesay that a convergence space is locally compact i f each neighborhood fi l ter containsa compac t se t . Le t /be a map f rom {S, q) onto (T, p) .(a) I f / i s p roper , then a subset A of Tis compact if and only if~^{A) is com pact. In

particular, inverse images of singletons are compact.(b) If (T , P) is locally comp act an d Hausdorf" , t h e n / is prop er if and only i f /~ ^( )is compact in S whenever A is compact in T.

(c) If (5, q) i s compac t and ( , p) is Hausdorff*, t h e n / i s p r o p e r .(d) If (S, q) is Hausdorflf a nd / is pro per , then (T, p) is Hausdorff*.(e) If / is a proper convergence q uotien t map an d (5 , q) is -regular (see [4]) for

a g iven card inal number X, then (T, p) i s a lso - regular.

4. E X T E N S I O N S T O T H E S T O N E - C E C H C O M P A C T I F I C A T I O NI t is shown in [5] that every Hausdorff convergence space (5, q) can be embedded

in a com pa ct H ausdorfF space (S *, ^*) where S* is a set of u.f. 's from S; this spac e,along with the natural embedding x - x is called the Stone-Cech compactificationof (S, q). Th is term inolog y is justified by th e fact th at a m ap / : (S, q) -^ ( , p) hasa u niq ue con tinu ous extension / * : (S* , g*) -> (T, p) if the range space is regula r,Hausdorff, and c om pact. Inde ed, / * is well-defined whenever (T, p) is compact andHausdorff; regularity is needed to secure continuity. The reader is referred to [5] forthe details of this construction.

21


9/10

Lemma 4 . 1 . Let f : {S, q) -> ( , p) be a map, with ( , p) compact and Hausdorjf.Iff^ is continuous, thenf^{rq*{A)) = {/{ )/ all A a S.

Proof. Since / * is con tinuous , / *( ^* ( ) ) {/{ ) If e {/{ )), then thereis a u.f. p-converging to such that/(y4) is in ; thus each member o^^{ )intersects A. Let Jf be a u . f . f iner than/"^( j f ) which contains A. If ^-convergesto some X in S, then x e ^{ ) and /* ( ^ *( )) . If does not ^-converge, then(in the notation of [5]) [ J T ] ^*-converges to and Am implies e ^*( ).Fu r thermore , { ) = , im ply in g /* ( * ( )) .

Proposition 4.2. Lef / : (S , g) -> (T , jp) be a convergence quotient map with (S, q)regular and (T , p) compact Hausdorff, Thenf^ is a convergence quotient map if andonly if{T, p) is regular.

Proof. Assume that /* i s a convergence quot ien t map and le t ^ p-converge to y.T hen there i s J^ (r , p) is a prop er m ap and ( , p) is com pact, then {S, q) is necessarilycompact. In this case we have the tr ivial results (S*, q^ ) = (S, q) a n d / * = / .

In the next example we show that an open m a p / c a n have an ex te nsi on /* which i snot neighborhood-preserving and hence not open.Example 4.3 . Let S = (0, 2] , T = [0, 1] be intervals on the rea l Une, with q and p

the usual topologies on S an d T , respectively. Define / : S -> T as follows: / ( x ) == 1. X fo r X in (0 , l ] ; / ( x ) = x 1 for x in [1 , 2 ] . C lea r ly / i s an open map f rom Sonto T . Let Af^ = {1/n, l/(n + 1), . . . } and let Jf SL u.f. finer than the filter base[A n : n = 1, 2, . . . } . Since J does not ^-converge, J G S'^ - S and /* ( j ^ ) == l im/( j^) = 1 . Moreover , each e S'^ S must converge to 0 in the usualsense wi th l i m / ( j f ) = 1. Hence /* ( ' ') does no t be long to Vp{l) and thus/*(t ;^*(j^))is not equal to Vp{i).

Definition 4.4. A m a p / : {S, q) -> ( , p ) is almost open if, for each x in T, there is i n / " ^ ( x ) s u c h thsit f(vq{y)) = Vp{x).

An open map is almost open; for pretopological spaces, almost open mapscoincide with convergence quotient maps.

T he proof of our concluding propo sit ion is straightforward .Proposition 4.5. / / / : (S, q) -^ (T , p ) is almost open and ( , p ) is compact, Haus

dorjf, and regular, then f^ is almost open.22


10/10

References[1] N. Bourbaki, General Topology, Addison-Wesley Pub. Co. , Inc. (1966).[2] D. C. Kent, "Convergence Quot ient Maps", Fund. Math. , 65 (1969) 197-205 .[3] D. C. Kent and A. M. Carstens, '*A Note on Products of Convergence Spaces", Math. Ann. ,

182 ( 1 9 6 9 ) 4 0 - 4 4 .[4] D. C. Kent and G. D. Richardson, '^Minimal Convergence Spaces", (To appear) Trans. Amer.

Math. Soc .[5] G. D. Richardson, "A Stone-Cech Compactificat ion for Limit Spaces", Proc. Amer. Math.

Soc , 25 (1970) , 403-404 .Authors' addresses: Darre l l C . K en t , Dep artme ntof Mathemat ics Washington Sta te Univers ity ,

Pul lman, Washington 99163, U.S.A. , Gary D. Ri c h a r d s o n , Dep artment of Mathemat ics Eas tCarolina Universi ty, Greenvil le , North Carolina 27834, U.S.A.

23

Open and Proper Maps Between Convergence Spaces

Documents

Transcript of Open and Proper Maps Between Convergence Spaces