On the weight and pseudoweight of linearly ordered topological spaces

Shorter Notes: On the Weight and Pseudoweight of Linearly Ordered Topological

Spaces

Klaas Pieter Hart

Proceedings of the American Mathematical Society, Vol. 82, No. 3. (Jul., 1981), pp. 501-502.

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http://links.jstor.org/sici?sici=0002-9939%28198107%2982%3A3%3C501%3ASNOTWA%3E2.0.CO%3B2-B

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PROCEEDINGS OF THE

AMERICAN MATHEMATICAL SOCIETY Volume 82, Number 3, July 1981

SHORTER NOTES

The purpose of this department is to publish very short papers of unusually elegant and polished character, for which there is no other outlet.

ON THE WEIGHT AND PSEUDOWEIGHT OF LINEARLY ORDERED TOPOLOGICAL SPACES

KLAAS PIETER HART

Ass-. We derive a simple formula for the weight of a LOTS using the pseudoweight. As an application we give a very short proof of the nonorderability of the Sorgenfrey-line.

1.Definitions. Let (X, 7) be a TI-space.

A collection

G21

E 7 is called a +-base for X [3]if

(i)

G21

covers X, and

(ii)

n

{Ulx E U E %) = {x), for all x E X.

We put as usual +w(X) = min(lc111

1%

is a +-base for X). Recall that

c(X) = sup{[

G211

I

G21

c

7 and

G21

is disjoint),

and

w(X) = min{(%(

1%

c

7 and % is a base for X ) .

THEOREM.If X is a Linearly Ordered Topological Space (LOTS), then w(X) = c(X)

.

+w(X).

PROOF."

>

"is obvious.

"<

".Let

G21

= {U,ji,, bea+-baseforxwith = +!w(X).Foreachi E Zlet

{Cij)j,4 be the collection of convex components of

U,.

Put $8 = {Cijlj E Ji,

i E I ) . Since lJil

<

c(X) for all i, we see that 1%

I

<

c(X)

.

+v(X). We claim that % is a subbase for X.

Indeed, take x E X and (a, b) 3 x. Since fl{Blx E B E % ) = {x), there exist Ba, Bb E 3 such that x E Ba $ a and x E Bb $ b. Then x E Ba

n

Bb

c

(a, b).

Received by the editors September 24, 1980.

1980 Mathematics Subject Classifcation Primary 54A25, 54F05.

Key word and phrases. LOTS,Sorgenfrey-line, Michael-line, Souslin-line, weight, pseudoweight, #-base.

O 1981 American Mathematical Society 0002-993918l/OOOO-0339/$01.50

502 K . P . HART

3. Examples. We shall show that our theorem cannot be improved.

3.1. The Sorgenfrey-line S shows that we cannot replace "X is a LOTS" by "X is a GO-space". Indeed w(S) = 2", c(S) = w and @(S) = w [take all intervals with rational endpoints].

This shows once again that S is not a LOTS. For other, more involved, proofs, see for example [I] and [4].

3.2. The largest "natural" invariant below c(X) (in the case of LOTS) is I(X), the Lindelof number of X. We shall see that we cannot replace c by I:

Let A

c

R be a subset of cardinality 2" with the property that for any closed set C

c

R with either C

c

A or C

c

R \ A we have that C is countable.

Build a Michael-line M(A) by isolating every a E A. The resulting space is Lindelof [6]. Now let X be the associated LOTS of the GO-space M(A) [S].

X = { ( x , n) E R x Zlx E R \ A

+

n = 0) endowed with the lexicographic order. The map

f:

X + M(A) defined by f((x, n)) = x is a retraction with countable fibers; hence X is Lindelof.

Let us put Uq = {x E Xlx

<

(q, 0)), Vq = { x E Xlx

>

(q, 0)) and 0, = {(a, n)la E A). Then %. = {Uq)q,Q

u

{Vq)q,Q

u

{On),,, is a countable

+-base for X. Finally we have w(X) = IAI = 2".

3.3. The largest invariant-to our knowledge-below @(X) is psw(X) = min{ord(%)l %. is a +-base for X ) [2]. If we let Z be a dense left-separated subspace of a connected Souslin-line, then Z is a LOTS with a point-countable base

[7].

Thus we have w(Z) = w,

>

w = c(Z) .psw(Z); hence @ cannot be replaced by psw.

1. P. v. Emde Boas et al., Cardinal functions on ordered spaces, rapport ZN 33/70, Math. Centre, Amsterdam, 1970.

2. R. E. Hodel, Extensions of metrizaion theorems to higher cardinali@, Fund. Math. 87 (1975), 2 19-229.

3. I . Juhasz, Cardinal functions in topology-ten years later, Math. Centre Tract No. 123, Amsterdam,

1980.

4. D. J. Lutzer, A metrization theorem for linear& ordered spaces, Proc. Amer. Math. Soc. 22 (1%9), 557-558.

5. ,On generalized ordered spaces, Dissertationes Math. 89(1971).

6 . E. A. Michael, The product of a normal space and a metric space need not be normal, Bull. Amer. Math. Soc. 69 (1%3), 375-376.

7 . F. D. Tall, On the existence of non-metrizable hereditmM& Lindelof spaces with point-countable bases,

You have printed the following article:

Shorter Notes: On the Weight and Pseudoweight of Linearly Ordered Topological Spaces

Klaas Pieter Hart

Proceedings of the American Mathematical Society, Vol. 82, No. 3. (Jul., 1981), pp. 501-502.

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http://links.jstor.org/sici?sici=0002-9939%28198107%2982%3A3%3C501%3ASNOTWA%3E2.0.CO%3B2-B

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References

4

Shorter Notes: A Metrization Theorem for Linearly Orderable Spaces

David J. Lutzer

Proceedings of the American Mathematical Society, Vol. 22, No. 2. (Aug., 1969), pp. 557-558.

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