AlasTamarizAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 9, No. 1, 2008

pp. 67-76

The Čech number of Cp(X) when X is an
ordinal space

Ofelia T. Alas and Ángel Tamariz-Mascarúa
∗

Abstract. The Čech number of a space Z, Č(Z), is the pseudochar-
acter of Z in βZ. In this article we obtain, in ZF C and assuming SCH,
some upper and lower bounds of the Čech number of spaces Cp(X) of
realvalued continuous functions defined on an ordinal space X with the
pointwise convergence topology.

2000 AMS Classification: 54C35, 54A25, 54F05

Keywords: Spaces of continuous functions, topology of pointwise conver-
gence, Čech number, ordinal space

1. Notations and Basic results

In this article, every space X is a Tychonoff space. The symbols ω (or N), R,
I, Q and P stand for the set of natural numbers, the real numbers, the closed
interval [0, 1], the rational numbers and the irrational numbers, respectively.
Given two spaces X and Y , we denote by C(X, Y ) the set of all continuous
functions from X to Y , and Cp(X, Y ) stands for C(X, Y ) equipped with the
topology of pointwise convergence, that is, the topology in C(X, Y ) of subspace
of the Tychonoff product Y X . The space Cp(X, R) is denoted by Cp(X). The
restriction of a function f with domain X to A ⊂ X is denoted by f ↾ A. For
a space X, βX is its Stone-Čech compactification.

Recall that for X ⊂ Y , the pseudocharacter of X in Y is defined as

Ψ(X, Y ) = min{|U| : U is a family of open sets in Y and X =
⋂

U}.

Definition 1.1.

(1) The Čech number of a space Z is Č(Z) = Ψ(Z, βZ).
(2) The k-covering number of a space Z is kcov(Z) = min{|K| : K is a

compact cover of Z}.

∗Research supported by Fapesp, CONACyT and UNAM.



68 O. T. Alas and Á. Tamariz-Mascarúa

We have that (see Section 1 in [8]): Č(Z) = 1 if and only if Z is locally
compact; Č(Z) ≤ ω if and only if Z is Čech-complete; Č(Z) = kcov(βZ \ Z);
if Y is a closed subset of Z, then kcov(Y ) ≤ kcov(Z) and Č(Y ) ≤ Č(Z);
if f : Z → Y is an onto continuous function, then kcov(Y ) ≤ kcov(Z); if
f : Z → Y is perfect and onto, then kcov(Y ) = kcov(Z) and Č(Y ) = Č(Z); if
bZ is a compactification of Z, then Č(Z) = Ψ(Z, bZ).

We know that Č(Cp(X)) ≤ ℵ0 if and only if X is countable and discrete

([7]), and Č(Cp(X, I)) ≤ ℵ0 if and only if X is discrete ([9]).
For a space X, ec(X) (the essential cardinality of X) is the smallest car-

dinality of a closed and open subspace Y of X such that X \ Y is discrete.
Observe that, for such a subspace Y of X, Č(Cp(X, I)) = Č(Cp(Y, I)). In [8] it

was pointed out that ec(X) ≤ Č(Cp(X, I)) and Č(Cp(X)) = |X| · Č(Cp(X, I))

always hold. So, if X is discrete, Č(Cp(X)) = |X|, and if |X| = ec(X),

Č(Cp(X)) = Č(Cp(X, I)).
Consider in the set of functions from ω to ω, ωω, the partial order ≤∗ defined

by f ≤∗ g if f (n) ≤ g(n) for all but finitely many n ∈ ω. A collection D of
(ωω, ≤∗) is dominating if for every h ∈ ωω there is f ∈ D such that h ≤∗ f .
As usual, we denote by d the cardinal number min{|D| : D is a dominating
subset of ωω}. It is known that d = kcov(P) (see [3]); so d = Č(Q). Moreover,
ω1 ≤ d ≤ c, where c denotes the cardinality of R.

We will denote a cardinal number τ with the discrete topology simply as τ ;
so, the space τ κ is the Tychonoff product of κ copies of the discrete space τ .
The cardinal number τ with the order topology will be symbolized by [0, τ ).

In this article we will obtain some upper and lower bounds of Č(Cp(X, I))
when X is an ordinal space; so this article continues the efforts made in [1]
and [8] in order to clarify the behavior of the number Č(Cp(X, I)) for several
classes of spaces X.

For notions and concepts not defined here the reader can consult [2] and [4].

2. The Čech number of Cp(X) when X is an ordinal space

For an ordinal number α, let us denote by [0, α) and [0, α] the set of ordinals
< α and the set of ordinals ≤ α, respectively, with its order topology. Observe
that for every ordinal number α ≤ ω, [0, α) is a discrete space, so, in this case,
Č(Cp([0, α), I)) = 1. If ω < α < ω1, then [0, α) is a countable metrizable space,

hence, by Theorem 7.4 in [1], Č(Cp([0, α), I)) = d. We will analyze the number

Č(Cp([0, α), I)) for an arbitrary ordinal number α.
We are going to use the following symbols:

Notations 2.1. For each n < ω, we will denote as En the collection of intervals

[0, 1/2n+1), (1/2n+2, 3/2n+2), (1/2n+1, 2/2n+1), (3/2n+2, 5/2n+2), ...

..., ((2n+2 − 2)/2n+2, (2n+2 − 1)/2n+2), ((2n+1 − 1)/2n+1, 1].

Observe that En is an irreducible open cover of [0, 1] and each element in En
has diameter = 1/2n+1. For a set S and a point y ∈ S, we will use the symbol
[yS]<ω in order to denote the collection of finite subsets of S containing y.



The Čech number of Cp(X) when X is an ordinal space 69

Moreover, if γ and α are ordinal numbers with γ ≤ α, [γ, α] is the set of
ordinal numbers λ which satisfy γ ≤ λ ≤ α. The expression α0 < α1 < ... <
αn < ... ր γ will mean that the sequence (αn)n<ω of ordinal numbers is strictly
increasing and converges to γ.

Lemma 2.2. Let γ be an ordinal number such that there is ω < α0 < α1 <
... < αn < ... ր γ. Then Č(Cp([0, γ], I) ≤ Č(Cp([0, γ), I) · kcov(|γ|

ω).

Proof. For m < ω, F ∈ [γ[αm, γ]]
<ω = {M ⊂ [αm, γ] : |M| < ℵ0 and γ ∈ M}

and n < ω, define

B(m, F, n) =
⋃

E∈En

B(m, F, E)

where B(m, F, E) =
∏

x∈[0,γ] Jx with Jx = E if x ∈ F , and Jx = I otherwise.

(So, B(m, F, n) is open in I[0,γ].) Define

B(m, n) =
⋂

{B(m, F, n) : F ∈ [γ[αm, γ]]
<ω}.

Observe that B(m, n) is the intersection of at most |γ| open sets B(m, F, n).
Define G(n) =

⋃

m<ω
B(m, n), and G =

⋂

n<ω
G(n).

Claim: G is the set of all functions g : [0, γ] → [0, 1] which are continuous at
γ.

Proof of the claim: Let g : [0, γ] → [0, 1] be continuous at γ. Given n < ω there
is E ∈ En such that g(γ) ∈ E. Since g is continuous at γ, there is β < γ so that
g(t) ∈ E if t ∈ [β, γ]. Fix m < ω so that β < αm. For every F ∈ [γ[αm, γ]]

<ω

we have that g ∈ B(m, F, E) ⊂ B(m, F, n); hence, g ∈ B(m, n) ⊂ G(n). We
conclude that g belongs to G.

Now, let h ∈ G. We are going to prove that h is continuous at γ. Assume the
contrary, that is, there exist ǫ > 0 and a sequence t0 < t1 < ... < tn < ... ր γ
such that

(1) |f (tj ) − f (γ)| ≥ ǫ,

for every j < ω. Fix n < ω such that 1/2n+1 < ǫ.
Since h ∈ G, then h ∈ G(n) and there is m ≥ 0 such that h ∈ B(m, n).

Choose tnp > αm and take F = {tnp , γ}. Thus h ∈ B(m, F, n), but if E ∈ En
and h(γ) ∈ E, then h(tnp ) 6∈ E, which is a contradiction. So, the claim has
been proved.

Now, we have I[0,γ] \ G =
⋃

n<ω
(I[0,γ] \ G(n)), and

I[0,γ] \ G(n) =
⋂

m<ω

⋃

F ∈γ[αm,γ]ω

(I[0,γ] \ B(m, F, n)).

So I[0,γ] \ G(n)) is an F|γ|δ-set. By Corollary 3.4 in [8], kcov(I
[0,γ] \ G(n)) ≤

kcov(|γ|ω). Hence, Č(G) = kcov(I[0,γ] \ G) ≤ ℵ0 · kcov(|γ|
ω). Thus, it follows

that

Č(Cp([0, γ], I) ≤ Č(Cp([0, γ), I) · kcov(|γ|
ω ).

�



70 O. T. Alas and Á. Tamariz-Mascarúa

Lemma 2.3. If γ < α, then Č(Cp([0, γ), I)) ≤ Č(Cp([0, α), I)).

Proof. First case: γ = β + 1.
In this case, [0, γ) = [0, β] and the function φ : [0, α) → [0, β] defined by

φ(x) = x if x ≤ β and φ(x) = β if x > β is a quotient. So, φ# : Cp([0, β], I) →
Cp([0, α), I) defined by φ

#(f ) = f ◦φ, is a homeomorphism between Cp([0, β], I)
and a closed subset of Cp([0, α), I) (see [2], pages 13,14). Then, in this case,

Č(Cp([0, γ), I)) ≤ Č(Cp([0, α), I)).
Now, in order to finish the proof of this Lemma, it is enough to show that

for every limit ordinal number α, Č(Cp([0, α), I)) ≤ Č(Cp([0, α], I)).
Let κ = cof (α), and α0 < α1 < ... < αλ < ... ր α with λ < κ.

For each of these λ, we know, because of the proof of the first case, that
κλ = Č(Cp([0, αλ], I)) ≤ Č(Cp([0, α], I)). Let, for each λ < κ, {V

λ
ξ : ξ < κλ}

be a collection of open subsets of I[0,αλ] such that Cp([0, αλ], I) =
⋂

ξ<κλ
V λξ .

For each λ < κ and each ξ < κλ, we take W
λ
ξ = V

λ
ξ × I

(αλ,α). Each W λξ is open

in I[0,α) and
⋂

λ<κ

⋂

ξ<κλ
W λξ = Cp([0, α), I)). Therefore, Č(Cp([0, α), I)) ≤

κ · sup{κλ : λ < κ} ≤ κ · Č(Cp([0, α], I)). But κ ≤ |α| = ec([0, α]) ≤

Č(Cp([0, α], I)).

Then, Č(Cp([0, α), I)) ≤ Č(Cp([0, α], I)). �

Lemma 2.4. Let α be a limit ordinal number > ω. Then

Č(Cp([0, α), I)) = |α| · supγ<αČ(Cp([0, γ), I)).

In particular, Č(Cp([0, α), I)) = supγ<αČ(Cp([0, γ), I)) if cof (α) < α.

Proof. By Lemma 2.3, supγ<αČ(Cp([0, γ), I)) ≤ Č(Cp([0, α), I)), and, by Corol-

lary 4.8 in [8], |α| ≤ Č(Cp([0, α), I)).

For each γ < α, we write κγ instead of Č(Cp([0, γ), I)). Let {V
γ
λ

: λ < κγ}
be a collection of open sets in Iγ such that Cp([0, γ), I) =

⋂

λ<κγ
V

γ
λ

. Now

we put W
γ
λ

= V
γ
λ

× I[γ,α)]. We have that W
γ
λ

is open for every γ < α and

every λ < γ, and Cp([0, α), I) =
⋂

γ<α

⋂

λ<κγ
W

γ
λ
. So, Č(Cp([0, α), I)) =

|α| · supγ<αČ(Cp([0, γ), I)). �

In order to prove the following result it is enough to mimic the prove of
5.12.(c) in [5].

Lemma 2.5. If α is an ordinal number with cof (α) > ω and f ∈ Cp([0, α), I)),
then there is γ0 < α for which f ↾ [γ0, α) is a constant function.

Lemma 2.6. If α is an ordinal number with cofinality > ω, then Č(Cp([0, α], I)) =

Č(Cp([0, α), I)).

Proof. Let κ = Č(Cp([0, α), I)). There are open sets Vλ (λ < κ) in I
[0,α) such

that Cp([0, α), I) =
⋂

λ<κ
Vλ. For each λ < κ, we take Wλ = Vλ × I

{α}. Each

Wλ is open in I
[0,α] and

⋂

λ<κ
Wλ = {f : [0, α] → I | f ↾ [0, α) ∈ Cp([0, α), I)}.

For each (γ, ξ, E) ∈ α×α×En, we take B(γ, ξ, E) =
∏

λ≤α Jλ where Jλ = E

if λ ∈ {ξ + γ, α}, and Jλ = I otherwise. Let B(γ, ξ, n) =
⋃

E∈En
B(γ, ξ, E).



The Čech number of Cp(X) when X is an ordinal space 71

Finally, we define B(γ) =
⋃

ξ<α
B(γ, ξ, n), which is an open subset of I[0,α].

We denote by M the set
⋂

λ<κ
Wλ ∩

⋂

γ<α
B(γ). We are going to prove that

Cp([0, α], I) = M .
Let f ∈ Cp([0, α], I). We know that f ∈

⋂

λ<κ
Wλ, so we only have to prove

that f ∈
⋂

γ<α
B(γ). For n < ω, there is E ∈ En such that f (α) ∈ E. Since

f ∈ C([0, α], I), there are γ0 < α and r0 ∈ I such that f (λ) = r0 if γ0 ≤ λ < α.
Let χ < α such that χ + γ ≥ γ0. Thus, f ∈ B(γ, χ, n) ⊂ B(γ). Therefore,
Cp([0, α], I) ⊂ M .

Take an element f of M . Since f ∈
⋂

λ<α
Wλ, f is continuous at every

γ < α, thus f ↾ [γ0, α) = r0 for a γ0 < α and an r0 ∈ I.
For each n < ω, and each γ ≥ γ0, f ∈ B(γ, ξ, n) for some ξ < α. Then,

|r0 − f (α)| = |f (γ + ξ) − f (α)| < 1/2
n. But, these relations hold for every

n. So, f (α) must be equal to r0, and this means that f is continuous at every
point.

Therefore, Č(Cp([0, α], I)) ≤ |α| · Č(Cp([0, α), I)). Since Č(Cp([0, α), I)) ≥

ec([0, α)) = |α|, Č(Cp([0, α], I)) ≤ Č(Cp([0, α), I)). Finally, Lemma 2.3 gives

us the inequality Č(Cp([0, α), I)) ≤ Č(Cp([0, α], I)). �

Theorem 2.7. For every ordinal number α > ω,

|α| · d ≤ Č(Cp([0, α), I)) ≤ kcov(|α|
ω ).

Proof. Because of Theorem 7.4 in [1], Corollary 4.8 in [8] and Lemma 2.3 above,
|α| · d ≤ Č(Cp([0, α), I)).

Now, if ω < α < ω1, we have that Č(Cp([0, α), I)) ≤ kcov(|α|
ω ) because of

Corollary 4.2 in [1].
We are going to finish the proof by induction. Assume that the inequality

Č(Cp([0, γ), I)) ≤ kcov(|γ|
ω ) holds for every ω < γ < α. By Lemma 2.4 and

inductive hypothesis, if α is a limit ordinal, then

Č(Cp([0, α), I)) ≤ |α| · supγ<αkcov(|γ|
ω) ≤ kcov(|α|ω ).

If α = γ0+2, then Č(Cp([0, α), I)) = Č(Cp([0, γ0+1), I)) ≤ kcov(|γ0+1|
ω) =

kcov(|α|ω ).
Now assume that α = γ0 + 1, γ0 is a limit and cof (γ0) = ω. We know by

Lemma 2.2 that Č(Cp([0, γ0 + 1), I)) ≤ Č(Cp([0, γ0), I) · kcov(|γ0|
ω). So, by

inductive hypothesis we obtain what is required.
The last possible case: α = γ0 + 1, γ0 is limit and cof (γ0) > ω.
By Lemma 2.6, we have Č(Cp([0, γ0 + 1), I)) = |α| · Č(Cp([0, γ0), I). By

inductive hypothesis, Č(Cp([0, γ0), I) ≤ kcov(|α|
ω ). Since |α| ≤ kcov(|α|ω ), we

conclude that Č(Cp([0, α), I)) ≤ kcov(|α|
ω ). �

As a consequence of Proposition 3.6 in [8] (see Proposition 2.11, below) and
the previous Theorem, we obtain:

Corollary 2.8. For an ordinal number ω < α < ωω, Č(Cp([0, α), I)) = |α| · d.



72 O. T. Alas and Á. Tamariz-Mascarúa

In particular, we have:

Corollary 2.9. Č(Cp([0, ω1), I)) = Č(Cp([0, ω1], I)) = d.

By using similar techniques to those used throughout this section we can
also prove the following result.

Corollary 2.10. For every ordinal number α > ω and every 1 ≤ n < ω,

|α| · d ≤ Č(Cp([0, α)
n, I)) ≤ kcov(|α|ω ).

For a generalized linearly ordered topological space X, χ(X) ≤ ec(X), so
χ(X) ≤ Č(Cp(X, I)), where χ(X) is the character of X. This is not the
case for every topological space, even if X is a countable EG-space, as was
pointed out by O. Okunev to the authors. Indeed, let X be a countable dense
subset of Cp(I). We have that χ(X) = χ(Cp(I)) = c and Č(Cp(X, I)) = d.
So, it is consistent with ZF C that there is a countable EG-space X with
χ(X) > Č(Cp(X, I)).

One is tempted to think that for every linearly ordered space X, the relation
Č(Cp(X, I)) ≤ kcov(χ(X)

ω) is plausible. But this illusion vanishes quickly; in
fact, when d < 2ω and X is the doble arrow, then X has countable character and
ec(X) = |X| = 2ω. Hence, Č(Cp(X, I)) ≥ 2

ω > d = kcov(χ(X)ω) (compare
with Theorem 2.7, above, and Corollary 7.7 in [1]).

In [8] the following was remarked:

Proposition 2.11.

(1) For every cardinal number ω ≤ τ < ωω, kcov(τ
ω ) = τ · d,

(2) for every cardinal τ ≥ λ, kcov((τ +)λ) = τ + · kcov(τ λ), and,
(3) if cf (τ ) > λ, then kcov(τ λ) = τ · sup{kcov(µλ) : µ < τ}.

Lemma 2.12. For every cardinal number κ with cof (κ) = ω, we have that
kcov(κω) > κ.

Proof. Let {Kλ : λ < κ} be a collection of compact subsets of κ
ω. Let α0 <

α1 < ... < αn < ... be an strictly increasing sequence of cardinal numbers
converging to κ. We are going to prove that

⋃

λ<κ Kλ is a proper subset of
κω. Denote by πn : κ

ω → κ the n-projection. Since πn is continuous and Kλ
is compact, πn(Kλ) is a compact subset of the discrete space κ, so, it is finite.
Thus, we have that |

⋃

λ<αn
πn(Kλ)| ≤ αn < κ for each n < ω. Hence, for every

n < ω, we can take ξn ∈ κ \
⋃

λ<αn
πn(Kλ). Consider the point ξ = (ξn)n<ω

of κω. We claim that ξ 6∈
⋃

λ<κ
Kλ. Indeed, assume that ξ ∈ Kλ0 . There is

n < ω such that λ0 < αn. So, ξn ∈
⋃

λ<αn
πn(Kλ) which is not possible. �

Recall that the Singular Cardinals Hypothesis (SCH) is the assertion:

For every singular cardinal number κ, if 2cof (κ) < κ, then κcof (κ) = κ+.

A proposition, apparently weaker than SCH, is: “for every cardinal number
κ with cof (κ) = ω, if 2ω < κ, then κω = κ+.” But this last assertion is
equivalent to SCH as was settled by Silver (see [6], Theorem 23).



The Čech number of Cp(X) when X is an ordinal space 73

Proposition 2.13. If we assume SCH and c ≤ (ωω)
+, and if τ is an infinite

cardinal number, then

(∗) kcov(τ ω ) =







τ · d if ω ≤ τ < ωω
τ if τ > ωω and cof (τ ) > ω
τ + if τ > ω and cof (τ ) = ω

Proof. Our proposition is true for every ω ≤ τ < ωω because of (1) in Propo-
sition 2.11.

Assume now that κ ≥ ωω and that (∗) holds for every τ < κ. We are going
to prove the assertion for κ.

Case 1: cof (κ) = ω. By Lemma 2.12, kcov(κω) > κ. On the other hand,
kcov(κω) ≤ κω.

First two subcases: Either c < ωω or κ > ωω. In both subcases, we can
apply SCH and conclude that kcov(κω) = κ+.

Third subcase: c = (ωω)
+ and κ = ωω. In this case we have kcov((ωω )

ω) ≤
(ωω)

ω ≤ cω = c = (ωω)
+. Moreover, by Lemma 2.12, (ωω)

+ ≤ kcov((ωω )
ω).

Therefore, kcov((ωω )
ω) = (ωω)

+.

Case 2: cof (κ) > ω. By Proposition 2.11 (3), kcov(κω) = κ · sup{kcov(µω) :
ω ≤ µ < κ}. By inductive hypothesis we have that for each µ < κ

(∗∗) kcov(µω ) =







µ · d if ω ≤ µ < ωω
µ if µ > ωω and cof (µ) > ω
µ+ if µ > ω and cof (µ) = ω

First subcase: κ is a limit cardinal. For every µ < κ, kcov(µω ) < κ (be-
cause of (∗∗) and because we assumed that κ > (ωω)

+ ≥ c ≥ d); and so
sup{kcov(µω) : µ < κ} = κ. Thus, kcov(κω ) = κ.

Second subcase: Assume now that κ = µ+0 . In this case, by Proposition 2.11,
kcov(κω) = κ·kcov(µω0 ). Because of (∗∗) and because µ0 ≥ ωω, kcov(µ0)

ω ≤ κ.
We conclude that kcov(κω) = κ. �

Proposition 2.14. Let κ be a cardinal number with cof (κ) = ω. Then

Č(Cp([0, κ], I)) > κ.

Proof. Let 0 = α0 < α1 < · · · < αn < . . . be a strictly increasing sequence of
cardinal numbers converging to κ. Assume that {Vλ : λ < κ} is a collection
of open sets in I[0,κ] which satisfies Cp([0, κ], I) ⊂

⋂

λ<κ
Vλ. We are going to

prove that
⋂

λ<κ
Vλ contains a function h : [0, κ] → I which is not continuous.

In order to construct h, we are going to define, by induction, the following
sequences:



74 O. T. Alas and Á. Tamariz-Mascarúa

(i) elements t0, . . . , tn, ... which belong to [0, κ] such that

(1) 0 = t0 < t1 < · · · < tn < . . . ,
(2) ti ≥ αi for each 0 ≤ i < ω,
(3) each ti is an isolated ordinal, and
(4) κ = lim(tn);

(ii) subsets G0, ..., Gn, ... ⊂ [0, κ] with |Gi| ≤ αi for every i < ω, and such that
each function which equals 0 in Gi and 1 in {t0, ..., ti} belongs to

⋂

λ<αi
Vλ for

every 0 ≤ i < ω and (
⋃

n
Gn) ∩ {t0, ..., tn, ...} = ∅;

(iii) functions f0, f1, ..., fn, ... such that f0 ≡ 0, and fi is the characteristic
function defined by {t0, ..., ti−1} for each 0 < i < ω.

Let f0 be the constant function equal to 0. Assume that we have already
defined t0, ..., ts−1, G0, ..., Gs−1 and f0, ..., fs−1. We now choose an isolated
point ts ∈ [αs, κ] \ G0 ∪ ... ∪ Gs−1 (this is possible because |G0 ∪ ... ∪ Gs−1| <
κ). Consider the characteristic function defined by {t0, ..., ts−1, ts}, fs. This
function is continuous, so it belongs to

⋂

λ<αs
Vλ. For each λ < αs, there is

a canonical open set Asλ of the form [fs; x
s
1, ..., x

s
ns(λ)

; 1/ms(λ)] = {f ∈ I[0,κ] :

|fs(x
s
i ) − f (x

s
i )| < 1/m

s(λ) ∀ 1 ≤ i ≤ ns(λ)} satisfying fs ∈ A
s
λ ⊂ Vλ. For

each λ < αs we take F
s
λ = {x

s
1, ..., x

s
ns(λ)

}. Put Gs =
⋃

λ<αs
F sλ \ {t0, ..., ts}.

It happens that {f ∈ I[0,κ] : f (x) = 0 ∀ x ∈ Gs and f (ti) = 1 ∀ 0 ≤ i ≤ s} is
a subset of

⋂

λ<αs
Vλ. This finishes the inductive construction of the required

sequences.
Now, consider the function h : [0, κ] → [0, 1] defined by h(x) = 0 if x 6∈

{t0, ..., tn, ...}, and h(tn) = 1 for every n < ω. This function h is not continuous
at κ because h(κ) = 0, κ = lim(tn), and h(tn) = 1 for all n < ω.

Now, take λ0 ∈ κ. There exists l < ω such that λ0 < αl. Since h is equal
to 0 in Gl and 1 in {t0, ..., tl}, then h ∈

⋂

λ<αl
Vλ. Therefore, h ∈ Vλ0 . So,

Cp([0, κ], I) is not equal to
⋂

λ<κ
Vλ. This means that Č(Cp([0, κ], I)) > κ. �

Theorem 2.15. SCH + c ≤ (ωω)
+ implies:

Č(Cp([0, α), I)) =







































1 if α ≤ ω
|α| · d if α > ω and ω ≤ |α| < ωω
|α| if |α| > ωω and cof (|α|) > ω
|α| if cof (|α|) = ω and α is a cardinal number > ωω
|α| if |α| = ωω and d < (ωω)

+

|α|+ if cof (|α|) = ω, |α| > ωω, α is not a cardinal number
|α|+ if |α| = ωω and d = (ωω)

+

Proof. If α ≤ ω, Cp([0, α), I) = I
[0,α), so Č(Cp([0, α), I)) = 1.

If α > ω and ω ≤ |α| < ωω, we obtain our result because of Theorem 2.7
and Proposition 2.13.

If |α| > ωω and cof (|α|) > ω, by Theorem 2.7 and Proposition 2.13,

|α| · d = |α| ≤ Č(Cp([0, α), I)) ≤ kcov(|α|
ω ) = |α|.



The Čech number of Cp(X) when X is an ordinal space 75

If cof (|α|) = ω and α is a cardinal number > ωω, by Lemma 2.4,

Č(Cp([0, α), I)) = |α| · supγ<αČ(Cp([0, γ), I)).

The number α is a limit ordinal and for every γ < α,

Č(Cp([0, γ), I)) ≤ |γ|
+ · d.

Since d ≤ (ωω)
+ < |α|, then Č(Cp([0, α), I)) = |α|.

By Lemma 2.4 and Theorem 2.7, if |α| = ωω, then

ωω · d ≤ Č(Cp([0, α), I)) = |α| · supγ<αČ(Cp([0, γ), I)) ≤ |α| · supγ<α(|γ|
+ · d).

Thus, if |α| = ωω and d < (ωω)
+, Č(Cp([0, α), I)) = |α|.

Assume now that cof (|α|) = ω, |α| > ωω and α is not a cardinal number.
There exists a cardinal number κ such that κ = |α| and [0, α) = [0, κ]⊕[κ+1, α).
So, Č(Cp([0, α), I)) = Č(Cp([0, κ], I)) · Č(Cp([κ + 1, α), I)) = Č(Cp([0, κ], I))
(see Proposition 1.10 in [8] and Lemma 2.3). By Theorem 2.7 and Proposition
2.14, κ · d ≤ Č(Cp([0, κ], I)) ≤ κ

+. Being κ a cardinal number > ωω with

cofinality ω, it must be > (ωω)
+; so κ > d and, then, κ ≤ Č(Cp([0, κ], I)) ≤ κ

+.

Now we use Proposition 2.14, and conclude that Č(Cp([0, α), I)) = κ
+ = |α|+.

Finally, assume that |α| = ωω and d = (ωω)
+. By Theorems 2.7 and Propo-

sition 2.13 we have

|α| · d ≤ Č(Cp([0, α), I)) ≤ kcov(|α|
ω ) = (ωω )

+.

And we conclude: Č(Cp([0, α), I)) = |α|
+. �

References

[1] O. T. Alas and Á. Tamariz-Mascarúa, On the C̆ech number of Cp(X), II Q & A in
General Topology 24 (2006), 31–49.

[2] A. V. Arkhangel’skii, Topological Function Spaces, Kluwer Academic Publishers, 1992.
[3] E. van Douwen, The integers and topology, in Handbook of Set-Theoretic Topology,

North-Holland, 1984, Amsterdam–New-York–Oxford, 111–167.
[4] R. Engelking, General topology, PWN, Warszawa, 1977.
[5] L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag, New-York-

Heidelberg-Berlin, 1976.
[6] T. Jech, Set Theory, Academic Press, New-York-San Francisco-London, 1978.
[7] D. J. Lutzer and R. A. McCoy, Category in function spaces, I, Pacific J. Math. 90

(1980), 145–168.

[8] O. Okunev and A. Tamariz-Mascarúa, On the C̆ech number of Cp(X), Topology Appl.
137 (2004), 237–249.

[9] V. V. Tkachuk, Decomposition of Cp(X) into a countable union of subspaces with “good”
properties implies “good” properties of Cp(X), Trans. Moscow Math. Soc. 55 (1994),
239–248.



76 O. T. Alas and Á. Tamariz-Mascarúa

Received August 2006

Accepted February 2007

Ofelia T. Alas (alas@ime.usp.br)
Universidade de São Paulo, Caixa Postal 66281, CEP 05311-970, São Paulo,
Brasil.

Ángel Tamariz-Mascarúa (atamariz@servidor.unam.mx)
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional
Autónoma de México, Ciudad Universitaria, México D.F. 04510, México.