An easy proof that βN—N —{p} is not normal

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JA N V A N M ILL *

AN EASY PROOF TH AT p N - N - { p } IS N O T NORMAL

Abstract. We give a simple proof that, under C H , P N —N —{p} is not norm al for any p c f i N —

One o f the m ost outstanding open problems in general topology is the question whether N * — {p) (for any space X we write X * = f l X—X ) is n o t norm al for any p e N *. U nder CH the question has been answered in the affirmative: for non / ’-points by Gillman (see [1]) and for P-points independently by Rajagopalan [8]

and W arren [10], The p ro o f o f the non P-point case uses Parovicenko’s [7] charac

terization of N * and the known proofs of the P-point case do n o t m ake use of this characterization. The aim of this note is to give a simple p ro o f th at N * — {/>'}

is no t norm al under CH. O ur p roo f is different since we use Parovicenko’s charac

terization in the P-point case and use the P-point case to solve the non P-point case.

It will be convenient to call a space X a Parovicenko space if

(a) X is a zero-dimensional compact space w ithout isolated points o f weight 2®, (b) every two disjoint open Fa's in X have disjoint closures,

(c) every nonem pty G

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in X has nonem pty interior.

Notice that (b) implies th at every countable subspace is C*-embedded.

It is known, [7], [3], th at C H is equivalent to the statem ent th at every Parovicenko space is hom eom orphic to N *.

The following lemma is known. It follows directly from the p ro o f of G illm an’s [5] result that, under CH, N * — {/>'} is not C*-embedded in N * for any p. Since I do no t know a reference for it I will give the easy proof.

LEM M A (CH). I f p e N * , then there is a Parovicenko space X <= N * containing p such that p is a P-point o f X.

P r o o f . W.l.o.g, p is no t a P-point, so take an open Fa U <=■ N * w ith p e U~ — U.

Let, by C H , {C„: a < t u j enum erate all nonem pty clopen subsets o f N * containing p.

By (b) and (c) we can find for each a < cot a nonem pty clopen Ea cz N * such that E x c n p<ac p ( U u yjfi<&Ep) .

Received April 25, 1983.

A M S (M OS) subject classification (1980). Prim ary 54G10.

* W iskunding Sem inarium Vrije U niversitat A m erdam , H olland.

« — Annales ₈₁

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If r = ( U « <«.,-£«) and if J e Y n ( U — {/>'}) then for some p < c o lt

y

^

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which condraticts (b). Hence F n t/~ = {p}. This implies th at p is a P -point o f Y for if Fez Y —{p} is any F„, then F n U~ = 0 and consequently, by (b), F~ n U ~ =

= 0 , i.e. p $ F~.

F o r each a < Mj take p x e E x and put X — [px : a < co{} ~ — {px : a < coj}.

Since p is a / ’-point of X and since {px : x < p) — {px : x < p} x N * for any co < p < cols it easily follows th at X satisfies (c). T hat X satisfies (b) is clear since (b) is closed hereditary in norm al spaces. This implies that A' is a Parovicenko space, since X clearly satisfies (a).

By the above Lemma we only need to show th at N * — {p'} is not norm al for any P -p o in tp e N *. Since W. R udin [9] showed that N * — {p} x N * — {q'} ifp , q e N * are P-points (under CH), the p ro o f is completed by the following

EX A M PLE 1. There is a Parovicenko space X having a P-point p such that X — {p} is not normal.

Let Z i = « j< , py e (©! + 1 ) x (co

1

+ 1 ) : p < x} and let Y = (cox Z)*. We claim that Y is a Parovicenko space. c a x Z is strongly zero-dimensional, hence so is P( c oxZ) , [6, 16.11]. Also, o x Z is a Lindelof space with weight cot , hence c o x Z has co® = 2® clopen subsets, hence /?(a> x Z ) has weight 2“ . It is clear th at Y has no isolated points. Y satisfies (b), since co x Z is a-com pact and locally compact, [6, 14.27]. Finally, Y satisfies (c), since co x Z is real com pact and locally compact, [4, 3.1].

Let n: c o x Z -» Z be the projection and pn its Stone extension.

CLAIM . P n~

1

((co1, &>!>) = (cox {<co1; cOi)'})- . Clearly (co x {<cox, coj)})” <= p n ~

1

((a)1, co!>). Take

xej87c"1« c o 1,co1» - ( c o x { < c o 1, CO!)'})- .

Let C be a clopen neighborhood o f x in P ( c o x Z ) which misses cox{<col5 coj)].

Then x e (C n (co x Z))~ and consequently

Pn(x)

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P n ( ( C n (co x Z) ) ~) <= (n (C n (co x Z)))~.

n ( C n ( c o x Z ) ) is c-com pact because C n ( c o x Z ) is cr-compact, and <co

1

,co1)$

$ n ( C n ( c o x Z)), and <coj, cox> £ (n ( C n (co x Z )))" , which is a contradiction since p n (x ) = <col5 coj>.

We conclude th at 7 - ( c o x {<co1; «!>'})* adm its a perfect m ap on to Z — {< « !, coj)'}, and hence is not norm al.

Let X = Y/(co x {<0)!, coj)'})* be the quotient space obtained from 7 by collapsing (cox{<cox, «>!>'})* to a point. We claim that X and p — { (o x {<col5 cot )'})*'} are as required. First, since <co1,co1> is a P -point o f Z, it easily follows th at (cox

x { (o jj, £»!>})“ is a P-set of P ( c o x Z ) (a subset A o f a space S is called a P-set whenever the intersection o f countably many neighborhoods o f A is again a neigh

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borhood of A), hence (co x {{w j, coj)})* is a jP-set of (coxZ )* and consequently, p is a P-point o f X. Second, X is a Parovicenko space. This follows from the fact th at Y is a Parovicenko space and th at (co x {<0^ , c«i>'})* is a nowhere dense closed P-set of Y.

The above example suggests the question whether it can be proved in Z FC that for any Parovicenko space X and for any p e X it is true th a t X — {p} is not norm al. U nfortunately, this is not possible, as the following example shows.

EX A M PLE 2. There is a compact zero-dimensional space X without isolated points o f weight a

>2

• 20J which satisfies (b) and (c), having a P-point p such that X — {p\

is both normal and C*-embedded in X.

Let P = {a ^ co

2

■ cf(a) ^ co/}. Put Y = fiP and X = Y —{ y e Y \ y is isolated'}.

Van Douwen [2] showed th at Y — {a>2} is almost compact, i.e. if A and B are disjoint closed subsets of Y — {co2} then one o f them is compact, and th at Y has weight o

>2

■ 2°\ T hat implies th at X — {a>2} is alm ost compact, and hence is norm al and C*-embedded in X. Since X has clearly weight tu2 • 2“ it remains to be shown th at X satisfies (b) and (c). T hat X satisfies (b) is trivial since Y satisfies (b) ([2]). Let G be any nonem pty closed Gs o f X. I f G n P ^ 0 , then int G ^ 0 , since P n X con

sists of P-points o f X. Hence a}- for certain a < co

2

■ By transfinite induction it is easy to show th at X n {£, e P : £ ^ a) “ has the property th at each nonem pty Gt has nonem pty interior. Since all these sets are clopen in X it follows that G has nonem pty interior.

Since the space of the above example is a Parovicenko space if CH fails our claim follows. It is interesting th at such a space exists since it shows th at the p ro perties (a), (b) and (c) of N * are not enough to prove th at N * — {p} is n o t norm al for any p e N * in Z FC alone.

Since, as rem arked earlier, under CH, N * — {p} is neither norm al nor C*- -embedded in N * , we have also obtained the following result.

TH EO R EM . Each o f the following statements is equivalent to C H :

(a) i f X is any Parovicenko space and p e X then X — {p} is not normal', (b) i f X is any Parovicenko space and p e X then X — {p} is not C*-embedded in X.

R E F E R E N C E S

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12] E. K . V A N D O U W E N , A basically disconnected normal space <P with = 1, Canad- J. M ath.

b ] E. K . VAN D O U W E N and J. V AN M ILL, Parovicenko's characterization o f fico —co implies C H , Proc. Amer. M ath. Soc. 72 (1978), 539—541.

[4] N . J. F IN E and L. G IL L M A N , Extension o f continuous functions in fiN, Bull. A mer. M ath.

Soc. 66 (1960), 376—381.

[5] L. G IL L M A N , The space fiN and the continuum hypothesis, General Topology and its relations to Modern Analysis and Algebra 2 (Proc. Second Praque Topological Sympos., 1966) (1967), 144— 146.

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[6] L. G IL L M A N an d M . JE R IS O N , Rings o f continuous functions, V an N ostrand 1960.

[7] 1.1. PA R O V IĆ E N K O j On a universal bicompact o f weight, D oki. A kad. N au k SSSR 150 (1963), 36—39 (Soviet M ath. D okl. 4 (1963), 592— 595).

[8] M . R A JA G O PA LA N , fiN—N —{ p} is not normal, J. Indian M ath. Soc. 36 (1972), 173— 176.

[9] W . R U D IN , Homogenity problems in the theory o f Ćech compactifications, D uke M ath. J. 23 (1956), 409—419.

[10] N . M . W A R R E N , Properties o f Stone-Ćech compactifications o f discrete spaces, Proc. Amer.

M ath. Soc. 33 (1972), 599— 606.