AbstractA new inductive technique is given for the construction of certain kinds of special points in the Čech-Stone compactification of X. This technique significantly simplifies the proofs of some known results on remote points and is also used to show that noncompact first-countable spaces without isolated points have far points

van Douwen, Eric K.

Chambre syndicale des artistes dramatiques, lyriques et musiciens. Auteur du texte

Gallica, Bibliotheque Numerique

La France théâtrale : organe de la Chambre syndicale des artistes dramatiques, lyriques et musiciens

Eric K. van Douwen

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P-remote points of X

Elsevier - Publisher Connector 

Topology and its Applications 5 I ( 1993) I4 1- I46 
North-Holland 
141 
van Douwen 
Abstract 
van Douwen, E.K., P-remote points of X, Topology and its Applications 5 1 ( I993 ) 14 l- 146. 
A new inductive technique is given for the construction of certain kinds of special points in the 
tech-Stone compactification of X. This technique significantly simplifies the proofs of some 
known results on remote points and is also used to show that noncompact first-countable spaces 
without isolated points have far points. 
Kqvwords: Remote points, n-linked collections, far points. 
1. introduction 
All spaces considered are Tychonoff. As usual, X* denotes PX -X. If A E X 
then Cl A denotes the closure of A in PX. A space X is called crowded if it has no 
isolated points (and is nonempty) and recall that a space is not pseudocompact if 
there is an infinite discrete collection “I’ of nonempty open sets. A collection 9 of 
sets is n-linked if n 9 # 0 for each 9 with 0 # 9 E 3 and 19 1 G n. 
For a property 9 (which may be intrinsic, like discrete, or extrinsic, like nowhere- 
dense), we call a point of X* a @remotn point of X if it is not in the pX-closure 
of any closed subspace of X that has 9. (So nowhere-dense-remote = remote, [2], 
and discrete-remote = far, [4].) Note that each nowhere-dense-remote point of a 
crowded space is discrete-remote. We call a P-shield for X any collection of closed 
subsets 9 of X such that for each closed P in X with 9 there is SE .Y with 
Cl SnCl P=(b. 
In [S] and independently in [l] n-linked P-shields were constructed simul- 
taneously for all n E o. The novel idea of this paper is to build them successively. 
This makes the construction very easy. As an application we prove that every normal 
nonpseudocompact space with an hereditarily closure-preserving r-base has a 
remote point and that nonpseudocompact first-countable crowded spaces have far 
points. 
0 166-864 l/93/$06.00 Q 1993~Elscvicr Science Publishers B.V. All rights rcscrvcd 
142 E. K. vun Douwen 
2. Building &shields successively 
Fact 2.1. A space X has a P-remote point p i$ it has a g-shield 9 which is centered 
(i.e., 0 3 ts0J&jSri;e 9 with (b# 3% 93 and free (i.e., 0 3=0). 
f. Given 9 we can find p E OS, :, Cl S. Clearly p E X * and p is @remote. Given 
p define 9’= (X n U: U is a closed neighborhood of p in PX}. Cl 
What makes Fact 2.1 useful is the idea from [4] to build centered collections 
from n-linked collections. Recall that a sequence {U,, : n E o} is discrete if it is 
pairwise disjoint and locally-finite. 
Fact 2.2. X has a LP-remote point if there is a discrete sequence (U,,),, of open sets in 
X and a sequence (Q,, of P-shields for X such that for each n E o, Z’,, is n-linked 
and satisjes (VL E Tf,)[ L,, c U,,]. 
Proof. It should be clear that 
Y= 
I 
U L,: mfoand L,,E.Z,, fornEw 
II --??I 
is a free centered collection of closed subsets of X. Also, for each closed P in X 
with 9 we can choose for each n E o an L,, E Y,, such that Cl P n Cl L,, = 0. Then, 
since the collection (L,, : n E w} is discrete, it follows that Cl P n Cl U,, L, = 0. Hence 
3’ is a g-shield. El 
In the previous proof we used the fact that discrete sequences are closure 
preserving in a strong sense. We introduce some well-known terminology and some 
natural generalizations for this setting. A collection 5%’ of subsets of a space X is 
called HCP (for hereditarily closure preserving) if & ( cl a(C) is closed in X 
whenever o(C) E C. In the context of PX, it is natural to introduce the following 
generalization: A collection % of subsets of a space X is called HC, P (for hereditarily 
C1 preserving) if whenever Cl U(C) c C is disjoint from Cl F for each C E 59, then 
Cl Uc,S %a( C) is disjoint from Cl F. Any locally finite family is HC,P. 
We call a collection % IHCP, or IHC, P, or ILF (I for initially) if ?Z admits a 
linear order < such that {A E %: A < C} is HCP, or HC, P, or locally finite, for each 
CE %. 
Proposition 2.3. Every countable collection is IHC,I? 
Proposition 2.4. (a) If X is normal then each IHCP collection of subsets of X is also 
IHC,I? 
(t?) Each ILF collection is IHC,P, 
(c) Each IHC,P collection is also IHCP. 
B-remote points qf X 143 
In [4J and later elsewhere, n-linked .9-shields were constructed simultaneously 
for all n E a. The novel idea of this paper is to build them successively. This makes 
the construction very easy. The crucial condition in the following lemma is (!). 
Lemma 2.5. For each space X and property 9 we have 
where 
( WP): for each nonempty open U in X there is an IHC,P collection W of nonempty 
open subsets of U, hence of X, such that 
( !) for each closed F in X with 9 there is W E W with F n W = 0. 
L,, : for each nonempty open Vin X there is an n-linked 9-shield ,cPfor X with u tP c V. 
Proof. Proof of Lo: {@} is a O-linked p-shield since every collection is O-linked. 
Proof that L, a L,,+, : Let V be any nonempty open subset of X, and let W be 
as in ( W9) for U = V. Fix the order < on W which witnesses that it is IHC, P. For 
each WE W, choose a nonempty open VW in X with Cl VW n Cl(X - W) =(d; this 
is possible since X is completely regular. Also for each WE 9%O, choose an n-linked 
Y-shield SW for X with U 9, c Vcrl. We claim that 
y= ClWU u s wp: WE W and Swl~Yw8 for W’< W 
W”. N’ 
is an (n + 1 )-linked g-shield for X. First of all, since W is IHC, P, 3 consists of 
closed sets. 
To see that Y is (n + l)-linked, consider any nonempty 5% 9 with 191 d n + 1. 
From the definition of 9 we see that we can define 
W=min{ W’E %? (~FE %)[cl Vwle F]), 
and also that F (=I z-1 VW E Yw u {cl VW} for all FE 9. Evidently F n cl VW = cl VMf 
for at least one FE 9. Since Yw is n-linked and since lJ Yu, c VW it follows that 
n 9~&,., Fncl VW #(d. Hence 3 is (n+l)-linked. 
Next consider any closed P in X with ,9. By ( !) there is WE W such that 
Pn W=@, and then Cl PnCl VW = 63. Also, for W’< W there is S+z .YNp, with 
Cl PnCl SW1 = c3. Then T = IJ w,,. cI, S,,,, u cl Vu, is a member of Y. Also, since I+’ 
is IHC,P, Cl P n Cl T = fl. Hence 3 is a Y-shield. Kl 
Combining Fact 2.2, Proposition 2.3 and Lemma 2.5, we immediately obtain the 
following. 
Corollary 2.6. Let X and 9 be as above and assume ( W9). 
(A) [f X is not pseudocompact then X has a .Y-remote point. 
( B) [f X is realcompact and not p: eudncompact then the set of 9-remote points of’ 
X is dense in X ‘. 
metrizabie spaces and no~ps~udocompact locally 
iserete-remote points, [4 
aces have discrete-remote points under 
ry n;oncompact Lindeliifspace has di 
wilt now use our results to identify another class of spaces having 
1. Because of Corollary 2.6 we want to show that if X is any first-countable 
crowded space, then 
( ): for each nonempty open U in X there is a countable collection 71’ of 
non<mpt> open subsets of U such that 
( !) no closed discrete set in X meets each member of 91: 
Givesa U pick any p E U, and let (II&)& be a neighborhood base at p such that 
B, G U and cl &, +, is a proper subset of Bk for k E W. Then clearly 74” = { Bk - cl Br, + I : 
k E o} satisfies the first part of W. Now consider any set F in X which meets each 
member of W. Then each neighborhood of p contains a point of F - { p}, hence F 
is not closed discrete. This proves i !). U 
Next, the technique is used to construct nowhere-dense-remote points (=remote 
points). A v-base for a space X is any collection 30 of nonempty open subsets of 
X such that for each nonempty open U in X there is a BE 93 with B C_ U. Observe 
that if X ia regular, then the collection of closures of members of a n-base is a 
nowhere-dense shield. 
y Proposition 2.4 and Lemma 2.5 we have established parts (a) and (b) of the 
wing theorem. 
3.2. JL3 nonpseudocompac? space X has a nowhere-dense-remote point lf an_, 
. . jiollo wing co Mditions hold : 
al atld has ati IfKP n-base; 
t b) X kiss an ILF r-base; or 
t 01 X has an IHCP w-base consisting of c/open sets. 
Notice that conditions (a) and (b) are satisfied by noncompact metrizable spaces 
and (a)-(c) are satisfied by P-spaces of n-weight wt. A space X is a P-space if 
every G,s is open. In order to prove (c) and another application motivated by the 
class of P-spaces we would like to weaken the ‘WC, P” hypothesis on ‘ttr in Lemma 
bviously we are simply interested in controlling the behaviour of the closure 
Without changing the proof of Lemma 2.5 we can weaken the hypotheses 
P-remote points af X 14s 
on IHC,P in two obvious ways. First of all we may restrict the possible choices of 
c(C). For example, for the proof of (c) we simply insist that o(C) is chosen to be 
a clopen set. However, for the inductive step analogue of Lemma 2.5, we have to 
be sure to construct Sshields consisting of clopen sets. 
Lemma 3.3. For a space X and each property 9 we have 
(09) * (Vn E d~)[R,,l, 
where 
(Og): for each nonempty open U in X there is an IHCP collection 0 of nonempty 
clopen subsets of U, hence of X, such that 
(!) for each closed F in X with 9 there is WE 0 with F n W = 8. 
R,, : for each nonempty open V in X there is an n-linked 9Qhield Y for X, consisting 
of clopen sets, with U 9 c V 
The other obvious place where the property IHC,P can be weakened is that it 
does not need to be applied to all closed sets F, only those which have property 9. 
That is, we cculd say that a collection % is HCP, or HC,P, with respect to 9, if 
cl U{n(C): C E %}ncl F =fl, or Cl U{cz( C): C E %} n Cl F = 0, whenever F has 
property 9 and o(C) c_ C is chosen so that cl U(C) n F, or Cl a(C) n Cl F, is empty 
for each C E %. For example, to illustrate this, we will consider P,-sets, i.e., a set 
such that every intersection of fewer than K neighborhoods is again a neighborhood. 
Therefore, every collection of size K is IHCP with respect to P,-sets. We can only 
get IHC,P with respect to sets whose closure in PX are P,-sets. However the two 
concepts coincide in compact spaces X which is our intended application. The 
obvious generalization has the same proof as Lemma 2.5. 
Lemma 3.4. For a space X and property 9 we have 
(Cg) * (VnE w)[L,,l, 
where 
(C9): for each nonempty open U in X there is a collection of open sets, w, which 
is IHC, P with respect o 9 such that 
( !) for each closed F in X with g there is WE 6 with F n W = 8. 
L, : for each nonempty open V in X there is an n-linked p-shield sPfor X with U 9 s V 
In addition, for the case of “nowhere-dense-P-sets”-remote points, Fact 2.2 can 
be strengthened by using the fact that countable collections are HCP with respect 
to P-sets. 
Fact 3.5. Let .p denote the collection qf nowhere dense P,-subsets ( K> cr) ) qf a corl?pact 
space A’. X has a centcrcd &shield (hence a point not included in any nowhc~e 
dense &set) if there is a pairwise disjoint collection ( U,,) ,, qf open sets in X and a 
scqwm ( Y&, qf .P-shields.for X strch that -for each new, Y,, is n-linked and satisfies 
W’~Z) [L,G CII. 
146 E.K. van Douwen 
Corollary 3.6 [3]. Let X be a compact space of v-weight =S K ( K > 0). Then there is 
an x E X suc~h that _Y ti K fur all nowhere-dense P,-sets K c X. 
References 
[ 11 S.B. Chae and .J.H. Smith, Remote points and G-spaces, Topology Appl. 11 (1980) 243-246. 
[2] N.J. Fine and L. Gillman, Remote points in /?lw, Proc. Amer. Math. Sot. 13 (1962) 29-36. 
[3] K. Kunen, J. van Mill and C.F. Mills, On nowhere dense closed F-sets, Proc. Amer. Math. Sot. 78 
(1980) 119-122. 
[43 E.K. van Douwen, Why certain tech-stone remainders are not homogeneous, Colloq. Math. 41 
(?979) 45-52. 
[ 51 E.K. van Douwen, Remote points, Dissertationes Math. 188 ( 1981) l-45. 
[6] E.K. van Douwen and J. van Mill, Spaces without remote points, Pacific J. Math. 105 (1983) 69-75. 
[7] J. van Mill, Weak P-points in tech-stone compactifications, Trans. Amer. Math. Sot. 273 (1982) 
657-678. 


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P-remote points of X

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