More on M. E. Rudin's Dowker space

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More on M. E. Rudin's Dowker Space

Klass Pieter Hart

Proceedings of the American Mathematical Society, Vol. 86, No. 3. (Nov., 1982), pp. 508-510.

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PROCEEDINGS OF THE

AMERICAN MATHEMATICALSOCIETY Volume 86, Number 3, November 1982

MORE ON M. E. RUDIN'S DOWKER SPACE

KLAAS PIETER HART

ABSTRACT.It is shown that M. E. Rudin's Dowker space is finitely-fully normal and orthocompact, thus answering questions of Mansfield and Scott.

0. Introduction. In [Ma] Mansfield defined the notions of K-full normality and finite-full normality. One of the questions he raised was, whether there exists a finitely-fully normal space which is not an a,-fully normal space.

In [Sc] Scott asked whether M. E. Rudin's Dowker space [Ru] is orthocompact. We answer both questions simultaneously by showing that the above-mentioned space is both finitely-fully normal and orthocompact. Mansfield's question is hereby answered since in [Ma] he showed that almost a,-fully normal spaces are countably paracompact. Almost K-full normality will not be defined here; it suffices to know that it is weaker than K-full normality.

1. Definitions and preliminaries.

1.0 K-full normality and orthocompactness. Let Y be a topological space,

"21

an open cover of Y and K 22 a cardinal. An open cover Y i s said to be a K-star (finite-star) refinement of %if for all

Y'

c

Ywith

I

?('

I

G K ( T ' finite) and fl

T'

# 0 there is a

U E

G2L

with

U

?r'

c

U , and ?(is a Q-refinement of

G2L

if ?(refines % and f l

Y'

is open for all ?p C

Y.

(Recent practice is to call Q-refinements interior-preserving open refinements.)

Y is called K-fully (finitely-fully) normal [Ma] if every open cover of Y has a K-star (finite-star) refinement. Y is called orthocompact [Sc] if every open cover of Y has a Q-refinement.

1.1 M. E. Rudin's Dowker space. Let F =

llr=,

(a,

+

1) endowed with the box topology. Furthermore let X' = { f E F: Vn E N cf( f(n)) > a,) and X = { f E X': 32' E N: Vn E N cf(f(n)) < a , ) . Then Xis M. E. Rudin's Dowker space [Ru].

We give an alternative description of the canonical base for X' (and X). For f , g E F w e s a y

f < gif f ( n ) < g ( n ) for all n, f G gif f ( n ) G g ( n ) for all n. F o r f , g E Fwithf ( g w e l e t

q,,=

{h E X ' : f c h s g )

Received by the editors February 9, 1982.

1980 Mathematics Subject Classification. Primary 54D20, 54620.

Key words and phrases. K-fully normal, finitely-fully normal, orthocompact.

01982 American Mathematical Society 0002-9939/82/OOOO-0212/$01.75

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~

LEMMA.

a. Every open cover of X' has a disjoint refinement consisting of basic open sets.

b. IfA, B C X are closed and disjoint then

CI,,A

n

C I ~ B= 0 . C] The next result is from [Ha].

2.1 LEMMA.For all n E N: (X')" is homeomorphic to X', and the homeomorphism can be chosen to map Xn onto X.

Now we are ready to prove the main result.

2.2 THEOREM.The space X is both 2-fully normal and orthocompact.

PROOF.Let"?LbeabasicopencoverofX.Put U = U{O X 0 X 0: 0 E %); U i s a neighborhood of {(x, x, x ) : x E X ) in x 3 . Using 2.1 and 2.0b find a neighborhood U' of {(x, x, x ) : x E X') in (X')3 such that U'

n

X3 = U.

For x E X' \ X, choose U, 3 x open such that U .

c

U'.

By 2.0a let

8'

be a disjoint basic open refinement of the open cover

~ ~ ' ~ ~

u

~{UXIX€X,fl.Y ~ ~ ~ O ~ ~

L e t 0 =

{O'

n

x:

0' E 0'). Let 0 E 0 and {x, y, z) C 0.

Then {x, y, z) C some V E % or {x, y, z ) C some

U,,

but then (x, y, z ) E

n

x3

C U, SO (x, y, z) E

v3

for some V E % in any case. This implies that {x, y, z)

C

v.

For each 0 E 0 define WO as follows: 0 = Up,, for some p , q E F, so put Wo =

{U,,,:

x E 0). Let W = U {Wo: 0 E 0 ) . Then W is both a 2-star and a Q-refinement of %.

First, assume Up,,

n

U,,, # 0 for some

U,,,

and U,,, in

W.

Then x and y are elements of the same 0 E 0 and hence p = q. Define p' by p'(n) =p ( n )

+

w , (n E N); then p <p'

<

x, y and p' E X, sop' E 0.

Pick u E "2t such that {p', x, y )

c

U. Since U is basic (and hence

<

-convex) and Up,z= {t: p'

<

t G Z ) for z = X, y, it follows that

U,,,

U Up,,

c

U. So

W

is a 2-star refinement of %. Second, let W' C W with f l

W'

# 0 . Then all W E W' are

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510 K . P.HART

contained in the same 0 E

8,

so

W'

=

{U,,,:

x E A) for some subset A of 0, where 0 = U,,,. Define f by f(n) = min{x(n): x E A). Then

n

W'

= Up,fis open. So

W

is a (2-refinement of %.

It now follows easily that X i s finitely-fully normal: 2.3 COROLLARY.X isfinitely-fully normal.

PROOF.Let % be an open cover of X. Let

lr,

be a 2-star refinement of %, and (inductively) let

Vn+,

be a 2-star refinement of

?I;,

(n E N). Since X is a P-space (G,'s are open) we can take the common refinement of all ?rn; call it

lr.

Let Y'

c

?r

be finite with

n

T

# 0 . Pick n E N such that 2" 21

T

I .

Since lrrefines

1T,

and since ?', is a 2"-star refinement of %, it follows that

U

'V'

is contained in some

U E % .

[Ha] K. P. Hart, Strong coilectionwise normalitv and M. E. Rudin's Dowker space, Proc. Amer. Math. SOC.83 (1981), 802-806.

[Ma] M. J. Mansfield, Some generalizations of full normality, Trans. Arner. Math. Soc. 86 (1957), 489-505.

[Ru] M. E. Rudin, A normal space Xfor which X X I is not normal, Fund. Math. 73 (1971), 179-186.

[Sc] B. M. Scott, Toward a product theory for orthocompacmess, Studies in Topology, Academic Press, New York, 1975, pp. 517-537.

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