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Applied General Topology

c© Universidad Politécnica de Valencia

Volume 13, no. 2, 2012

pp. 225-226

Correction to: Some results and examples
concerning Whyburn spaces

Ofelia T. Alas, Maira Madriz-Mendoza and Richard G. Wilson

Abstract

We correct the proof of Theorem 2.9 of the paper mentioned in the title
(published in Applied General Topology, 13 No.1 (2012), 11-19).

2010 MSC:Primary 54D99; Secondary 54A25; 54D10; 54G99

Keywords: Whyburn space, weakly Whyburn space, submaximal space,
scattered space, semiregular, feebly compact

There is an error in the proof of Theorem 2.9. A correct proof is as follows.

Theorem 2.9. If X is weakly Whyburn, then |X| ≤ d(X)t(X).

Proof. If X is finite, the result is trivial; thus we assume that X is infinite.
Suppose that d(X) = δ, t(X) = κ and D ⊆ X is a dense (proper) subset
of cardinality δ. Let D = D0 and define recursively an ascending chain of
subspaces {Dα : α < κ

+} as follows:
Suppose that for some α ∈ κ+ and for each β < α we have defined dense

sets Dβ such that |Dβ| ≤ δ
κ and Dγ ⊆ Dλ whenever γ < λ < α. If α is a

limit ordinal, then define Dα =
⋃
{Dβ : β < α} and then |Dα| ≤ |α|.δ

κ ≤
κ+.δκ = δκ. If on the other hand α = β + 1, and Dβ  X, then since X is
weakly Whyburn there is some x ∈ X \ Dβ and Bx ⊆ Dβ such that |Bx| ≤ κ,
cl(Bx) \ Dβ = {x}; thus necessarily, we have that |cl(Bx)| ≤ δ

κ and we define

Dα =
⋃

{cl(B) : B ⊆ Dβ, |B| ≤ κ, |cl(B)| ≤ δ
κ}.

Clearly Dα ! Dβ and since there are at most (δ
κ)κ such sets B it follows that

|Dα| ≤ δ
κ. If Dα = X for some α < κ

+, then we are done. If not, then we
define ∆ =

⋃
{Dα : α < κ

+}, and clearly |∆| ≤ κ+.δκ = δκ.



226 O. T. Alas, M. Madriz-Mendoza and R. G. Wilson

Thus to complete the proof it suffices to show that ∆ = X. Suppose to
the contrary; then, since ∆ is not closed and X is weakly Whyburn and has
tightness κ, there is some z ∈ X \ ∆ and some set B ⊆ ∆ of cardinality
at most κ, such that cl(B) \ ∆ = {z} and hence |cl(B)| ≤ δκ. Since the sets
{Dα : α < κ

+} form an ascending chain and cf(κ+) > κ, it follows that for some
γ < κ+, B ⊆

⋃
{Dα : α < γ} and hence z ∈ Dγ+1 ⊆ ∆, a contradiction. �

It should also be noted that Theorem 2.6 is not as claimed, an improvement
on the cited result of Bella, Costantini and Spadaro, since a Lindelöf P-space
is regular.

(Received October 2012 – Accepted October 2012)

O. T. Alas (alas@ime.usp.br)
Instituto de Matemática e Estat́ıstica, Universidade de São Paulo, Caixa Postal
66281, 05311-970 São Paulo, Brasil.

M. Madriz-Mendoza, R. G. Wilson (seber@xanum.uam.mx, rgw@xanum.uam.mx)
Departamento de Matemáticas, Universidad Autónoma Metropolitana, Unidad
Iztapalapa, Avenida San Rafael Atlixco, #186, Apartado Postal 55-532, 09340,
México, D.F., México.


	Correction to: Some results and examples concerning Whyburn spaces. By O. T. Alas, M. Madriz-Mendoza and R. G. Wilson