GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 6, 1995, 653-657

C(X) IN THE WEAK TOPOLOGY

N. V. VELIČKO

Abstract. Some relations between cardinal invariants of X and C(X) are established in the weak topology, where C(X) is the space of continuous real-valued functions on X in the compact-open topology.

Let X be a compact space. Denote by C(X) the space of continuous realvalued functions on X in the compact-open topology, by C′(X) ≡ (C(X))′ the vector space dual to C(X), i.e. the space of continuous linear forms on C(X), by C_{ω}(X) (C′_{ω}(X)) the space C(X) (C′(X)) in C′(X)-topology (C(X)-topology), and by C_{p}(X) the space C(X) in the topology of pointwise convergence.

Symbols |X|, ω, χ, d, πω, πχ, pω, nω denote the cardinality, weight, character, density, π-weight, π-character, pseudo-weight, and network weight, respectively (see, e.g., [1]).

In this paper we shall establish some relationship between cardinal invariants of X and C_{ω}(X).

Proposition 1. ω(C_{ω}(X)) ≤ exp ω(X).

Proof. Let ω(X) = τ. As is well known, ω(C(X)) = τ and |C(C(X))| ≤ exp τ. Since C′(X) ⊆ C_{p}(C(X)), we have |C′(X)| ≤ exp τ. Since C_{ω}(X) is a subspace of C_{p}(C′(X)), it follows that ω(C_{ω}(X)) ≤ ω(C_{p}(C′(X))) ≤ |C′(X)|. And finally, ω(C_{ω}(X)) ≤ exp ω(X), which completes the proof.

To get further estimates we need the following general proposition (σ(·, ·) stands below for weak topology [2]).

Proposition 2. Let E be a Banach space, E_{ω} = (E, σ(E, E′)), and S_{ω} be the unit closed ball in E_{ω}. Then d(E′) ≤ πχ(S_{ω}) ≤ πχ(E_{ω}) ≤ χ(E).

653 Proof. Let π = {V_{α} : α ∈ A} be a π-base at the point 0 ∈ S_{ω}. One may assume that |A| = πχ(S_{ω}) and V_{α} = S_{ω} ∩ {x ∈ E : |f_{αi}(x)| < ε_{α}, i ≤ n(α), f_{αi} ∈ E′}. Let Z′ be the linear hull of the set {f_{αi} : α ∈ A, i ≤ n(α)}. Prove that Z’ is dense in E′. Assume the contrary: there exists a form f ∈ E′\[Z′]. By the Hahn–Banach theorem there exists a linear form g ∈ E′′ such that g(f) = 1 and g(Z′) = 0. There is no loss of generality in assuming that g ∈ S′′. Let V = {x ∈ E : f(x) < 2^{−1}}. There exists α ∈ A such that S_{ω} ∩ V ⊆ V_{α}. The neighborhood

1991 Mathematics Subject Classification. 54C35.

Key words and phrases. Compact space, continuous real-valued functions, weak topology, cardinal invariants.

1072-947X/95/1100-0653$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

V (f_{αi}, f, ε_{α}, 2^{−1}, i ≤ n(α)) = {g′ ∈ E′′ : |g′(f_{αi})| < ε_{α}, |g′(f) − 1| < 2^{−1}}

of the point g in σ(E′′, E′)-topology of the space E′′ contains some element k(x), where x ∈ S_{ω} and k : E → E′ is a canonical embedding (this follows from Goldstine’s theorem [3]). But then x ∈ V_{α}\V, which contradicts the choice of V_{α}. We have proved that [Z′] = E′; it follows that α(E′) ≤ |Z′| = |A| ≤ πχ(S_{ω}).

Corollary 1. For E = C(X) the following is valid:

|X| ≤ d(C′(X)) ≤ πχ(S_{ω}).

Indeed, the canonical mapping q : q(t)(x) = x(t) transfers the set X onto a discrete subset of the space C′(X) (since |q(t) − q(t′)| = 2 for any points t, t′ from X). It follows that |X| ≤ d(C′(X)), and we can apply Proposition 2.

Question 1. Is it true that |X| = d(C′(X))?

One more assertion. Let F ⊆ E′. Set E_{F} = (E, σ(E, F)).

Proposition 3. χ(E_{F}) = ω(E_{F}).

Proof. Let {V_{α} : α ∈ A} be a fundamental family of σ(E, F)-neighborhoods of zero in E_{F} such that |A| ≤ χ(E_{F}). We can suppose that V_{α} = {x ∈ E : |f_{αi}(x)| < ε_{α}, f_{αi} ∈ F, i ≤ n(α)}. Fix some countable base π in ℝ. Put γ = {f_{αi}^{−1}(V) : α ∈ A, i ≤ n(α), V ∈ π}. Then |γ| = |A| and it suffices to show that γ is a subbase for the topology σ(E, F). Let x_{0} be an arbitrary point of E and let V be a σ(E, F)-neighborhood of x_{0}. The set V −x_{0} is σ(E, F)-neighborhood of zero, so that there exists α ∈ A such that V_{α} ⊆ V −x_{0}. Choose the sets W_{αi} ∈ π such that f_{αi}(x_{0}) ∈ W_{αi} ⊆ (f_{αi}(x_{0})− ε_{α}, f_{αi}(x_{0})+ε_{α}), i ≤ n(α). Set Γ = ∩{f_{αi}^{−1}(W_{αi}) : i ≤ n(α)} and prove that Γ ⊆ V. Let x ∈ Γ. Then f_{αi} ∈ W_{αi}. Hence either |f_{αi}(x) − f_{αi}(x_{0})| < ε_{α} or |f_{αi}(x − x_{0})| < ε_{α} for i ≤ n(α), and x − x_{0} ∈ V_{α} ⊆ V − x_{0} or x ∈ V. We have proved that γ is a subbase and so ω(E_{F}) ≤ |γ| = |A| ≤ χ(E_{F}).

C(X) IN THE WEAK TOPOLOGY 655

Summing up the above arguments, we arrive at the following Theorem 1.

ω(X) = d(C(X)) ≤ πχ(C_{ω}(X)) = πω(C_{ω}(X)) =
= χ(C_{ω}(X)) = ω(C_{o}m(X)) ≤ exp ω(X).

Proof. It is necessary here to employ Proposition 1, Corollary 1, Proposition 3, and the well-known fact that πχ = χ and πω = ω for every topological group.

Theorem 2.

|X| ≤ πχ(S_{ω}) = πω(S_{ω}) = χ(S_{ω}) = ω(S_{ω}) = d(C′(X)) ≤ exp ω(X).

Proof. Since d(C′(X)) ≤ ω(C′(X)) and nω(C_{p}(C′(X))) ≤ ω(C′(X)), we have nω(S′′_{ω}) ≤ d(C′(X)), where C′′_{ω} = ((C′_{ω}(X))′, σ(C′′(X), C′(X))) = (C′′(X), σ(C′′(X), C′(X))). As S′_{ω} is compact, ω(S′′_{ω}) = nω(S′′_{ω}). Since S_{ω} is embedded topologically in S′′_{ω}, ω(S_{ω}) ≤ ω(S′′_{ω}) = nω(S′′_{ω}) ≤ d(C′(X)). By Corollary 1, πχ(S_{ω}) ≥ d(C′(X)), so that πχ(S_{ω}) = πω(S_{ω}) = χ(S_{ω}) = ω(S_{ω}) = d(C′(X)).

Question 2. Is it true that ω(E_{ω}) ≥ ω(E)?

Question 3. Is it true that ω(E′) ≥ ω(E′_{ω})?

Note that the weight of C_{ω}(X) does not necessarily coincide with the weight of S_{ω}. Indeed, let X be a convergent sequence of real numbers. Then C(X) is isomorphic to the space C of all convergent sequences of real numbers. C′ = ℓ_{1} is separable, hence its unit closed ball is metrizable in C(X)-topology, i.e., χ(S_{ω}) = ω(S_{ω}) = ω_{0}. But C_{ω}(X) is nonmetrizable, so that χ(C_{ω}(X)) > ω(S_{ω}).

The problem of ψ-characters for C_{p}(X) was solved in [4]:

ψ(C_{p}(X)) ≤ d(X).  (1)

Since C_{ω}(X) maps onto C_{p}(X) one-to-one, from (1), it follows that

ψ(C_{ω}(X)) ≤ d(X).  (2)

Moreover, the following proposition is true.

Proposition 4. ψ(C_{ω}(X)) = pω(C_{ω}(X)) = d(C′_{ω}(X)).

Proof. Let {V_{α} : α ∈ A} be a subbase of C_{ω}(X) in zero having the least cardinality and consisting of standard sets V_{α} = {x ∈ C(X) : f_{αi}(x) < ε_{α}, i ≤ n(α)}. Let T be the C(X)-closure of the linear hull of the set {f_{αi} : α ∈ A, i ≤ n(α)}. Prove that T = C′(X). Suppose the contrary: there exists a point g ∈ C′(X)\T. By the theorem on separation of convex sets there exists a C(X)-continuous linear form f such that f(T) = 0 and f(y) > 0. By virtue of the C(X)-continuity, f ∈ C(X). For all αi we have f_{αi}(f) = 0.

Hence f ∈ V_{α} for all α, but ∩{V_{α} : α ∈ A} = {0}. We have a contradiction. Consequently, T = C′(X) and ψ(C_{ω}(X)) ≥ d(C′_{ω}).

Now let M = {g_{α} : α ∈ A} be a C′(X)-dense set of cardinality d(C′(X)). Then M is C(X)-separating family and the diagonal of mappings g_{α} produces a condensation of C_{ω}(X) in the product space ∏{R_{α} : α ∈ A}. From this it follows that pω(C_{ω}(X)) ≤ ω(∏{R_{α} : α ∈ A}) = |A|.

Proposition 4 underlies the following assertion.

Theorem 3. If X is an Eberline compactum, then ψ(C_{ω}(X)) = d(X).

Proof. Let Y ⊆ C_{p}(X) be an X-separating compactum. Then ω(Y) = d(Y) = d(X) = ω(X). One may suppose that Y lies in S_{ω}. By the Grothendieck theorem [5], Y is compact in C_{ω}(X). Then ω(γ) = pω(Y) ≤ pω(C_{ω}(X)) = ψ(C_{ω}(X)) or d(X) ≤ ψ(C_{ω}(X)). Reference to Proposition 4 completes the proof.

Proposition 5. The following statements are equivalent:

(1) C_{ω}(X) is a k-space;

(2) C_{ω}(X) is sequential;

(3) C_{ω}(X) is a Frechet–Urysohn space;

(4) C_{ω}(X) is metrizable;

(5) C_{ω}(X) is finite.

Proof. It suffices to show that (1)⇒(5).

Suppose that X is infinite. Then dim C(X) = ∞ and there exists a set A in C(X) such that 0 ∈ [A]_{ω}, where [A]_{ω} is the weak closure of A, but the intersection of A with any bounded set is finite (see, e.g., [6]).

Let K be an arbitrary compact set in C_{ω}(X). Then K is a Frechet– Urysohn space. Hence, if x ∈ [K ∩ A]_{ω}, there exists a sequence {x_{n}} of elements of A which converges to x. Any weak convergent sequence is always bounded. For the definition of A it follows that the set ∪{x_{n} : n ∈ ω} is finite, i.e., the sequence {x_{n}} is stationary: x_{n} ≡ x beginning with some n. Hence x ∈ A and K ∩ A is weakly closed. But this means that C_{ω}(X) is not a k-space.

The situation with S_{ω} is somehow different. Here there are other criteria for metrizability.

Proposition 6. S_{ω} is metrizable iff X is countable.

Proof. If S_{ω} is metrizable, then X is countable by Theorem 2. If S_{ω} is countable, then C_{p}(X) is metrizable. As X is compact, X is scattered. By the theorem from [7], S_{ω} is homeomorphic to S_{p}, hence S_{ω} is metrizable.

Then the following proposition is valid.

Proposition 7. If X is a scattered compactum, then S_{ω} has the Frechet– Urysohn (FU) property.

Proof. If X is scattered, then C_{p}(X) satisfies the FU-property. S_{ω} is homeomorphic to S_{p}, hence S_{ω} has the FU-property.

Consequently, if X is an uncountable scattered compactum, then S_{ω} is a nonmetrizable Frechet–Urysohn space.

In conclusion we shall prove the formula

d(C_{ω}(X)) = hd(C_{ω}(X)) = nω(C_{ω}(X)) = ω(X).  (3)

Proof. Obviously, d(Z) ≤ hd(Z) ≤ nω(Z) for any Z. But d(C_{p}(X)) = d(C(X)) = ω(X) [8], from which it follows that d(C_{ω}(X)) = ω(X). But nω(C_{ω}(X)) = ω(X), which completes the proof.

Aknowledgment

This research was supported by the Russian Fund for Fundamental Research, Grant 93-011-1401.

References

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4. J. Guthrie, Ascoly theorems and pseudocharacter of mapping spaces. Bull. Austral. Math. 10(1974), 403–408.

5. A. Grothendieck, Critères generaux de compaticité dans les espaces fonctionels généraux. Amer. J. Math. 74(1952), 165–186.

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8. N. Noble, The density character of function spaces. Proc. Amer. Math. Soc. 42(1974), 228–233.

(Received 18.10.1993)

Author’s address:

Institute of Mathematics and Mechanics

Ural Branch of the Russian Academy of Sciences

ul. Kovalevskoi 16, Ekaterinburg 620066

Russia