Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

ON THE METRIZABILITY OF SPACES WITH A SHARP BASE

CHRIS GOOD, ROBIN W. KNIGHT, AND ABDUL M. MOHAMAD

Abstract. A base B for a space X is said to be sharp if, whenever x ∈ X and (B_{n})_{n∈ω} is a sequence of pairwise distinct elements of B each containing x, the collection {⋂_{j≤n} B_{j} : n ∈ ω} is a local base at x. We answer questions raised by Alleche et al. and Arhangel’skiĭ et al. by showing that a pseudocompact Tychonoff space with a sharp base need not be metrizable and that the product of a space with a sharp base and [0, 1] need not have a sharp base. We prove various metrization theorems and provide a characterization along the lines of Ponomarev’s for point countable bases.

The notion of a uniform base was introduced by Alexandroff who proved that a space (by which we mean T_{1} topological space) is metrizable if and only if it has a uniform base and is collectionwise normal [1]. This result follows from Bing’s metrization theorem since a space has a uniform base if and only if it is metacompact and developable. Recently Alleche, Arhangel’skiĭ and Calbrix [2] introduced the notions of sharp base and weak development, which fit very naturally into the hierarchy of such strong base conditions including weakly uniform bases (introduced by Heath and Lindgren [11]) and point countable bases (see Figure 1 below). In this paper we look at the question of when a space, with a sharp base is metrizable. In particular, we show that a pseudocompact space with a sharp base need not be metrizable, but generalize various situations where a space with a sharp base is seen to be metrizable.

Definition 1. Let B be a base for a space X.

2000 Mathematics Subject Classification. 54E20, 54E30. Key words and phrases. Tychonoff space, pseudocompact, special bases, sharp base,

metrizability.

Reprinted from Topology and its Applications, in press, Chris Good, Robin W. Knight and Abdul M. Mohamad, On the metrizability of spaces with a sharp base, Copyright (2002), with permission from Elsevier Science [9].

Chris Good, Robin W. Knight and Abdul M. Mohamad, On the metrizability of spaces with a sharp base, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 125–134, Topology Atlas, Toronto, 2002; arXiv:math.GN/0204127.

(1) B is said to be sharp if, whenever x ∈ X and (B_{n})_{n∈ω} is a sequence of pairwise distinct elements of B each containing x, the collection {⋂_{j≤n} B_{j} : n ∈ ω} is a local base at x.

(2) B is said to be uniform if, whenever x ∈ X and (B_{n})_{n∈ω} is a sequence of pairwise distinct elements of B each containing x, then (B_{n})_{n∈ω} is a local base at x.

(3) B is said to be weakly uniform if, whenever B′ is an infinite subset of B, then ⋂ B′ contains at most one point.

(4) B is said to be a weak development if B = ⋃_{n∈ω} B_{n}, each B_{n} is a cover of X and, whenever x ∈ B_{n} ∈ B_{n}for each n ∈ ω, then {⋂_{j≤n} B_{j} : n ∈ ω} is a local base at x.

Arhangel’skiĭ et al. prove that a space with a sharp base has a point countable sharp base ([2] and [4]) and is metaLindelöf. Moreover a weakly developable space has a G_{δ}-diagonal and a submetacompact space with a base of countable order is developable [2].

We note in passing that the obvious definition of ‘uniform weak developability’ (having a base G = ⋃{G_{n} : n ∈ ω} such that each G_{n} is a cover and whenever x ∈ G_{n} ∈ G_{n}, {G_{n}}_{n} is a base at x) is simply a restatement of developability. We also note that a space with a σ-disjoint base need not have a sharp base: Bennett and Lutzer [7] construct a first countable (and a Lindelöf) example of a non-metrizable LOTS with σ-disjoint bases (and continuous separating families), which can not have a sharp base by Theorem 2.

[figure: development; uniform base ≡; metacompact+development; sharp base; ≡; point countable; sharp base; base of; countable order; weak; development; weakly; uniform base; point countable; base]

Figure 1

When is a space with a sharp base metrizable? We summarize the relevant results of [2], [4] and [6] in the following theorem.

Theorem 2. Let X be a regular space with a sharp base, then X is metrizable if any of the following hold:

(1) X is separable; (2) X is locally compact (so a manifold with sharp base is metrizable);

(3) X is countably compact;

(4) X is pseudocompact and CCC;

(5) X is a GO space.

A space is pseudocompact if every continuous real valued function is bounded. Every (Tychonoff) pseudocompact space with a uniform base is metrizable (see [19], [16] or [18]), whilst a pseudocompact space with a point-countable base need not be metrizable [17]. Moreover pseudocompact Tychonoff spaces with regular G_{δ}-diagonals are metrizable [14], whilst Mrowka’s Ψ space is an example of a pseudocompact, non-metrizable Moore space. So it is natural to ask (see [2] and [4]) whether every pseudocompact space with a sharp base is metrizable. The space P of Example 3 shows that the answer to this question is ‘no.’ In addition, P answers a number of other questions in the negative: Alleche et al. ask whether the product X × [0, 1] has a sharp base if X does; Heath and Lindgren [11] ask whether a space with a weakly uniform base has a G_{δ}^{∗}-diagonal; and P is another example (see [17] and [20]) of a pseudocompact space with a point countable base that is not compact, and is a non-compact pseudocompact space with a weakly uniform base, answering questions of Peregudov [15].

Example 3. There exists a Tychonoff, non-metrizable pseudocompact space with a sharp base but without a G_{δ}^{∗}-diagonal whose product with the closed unit interval does not have a sharp base.

Proof. Our example is a modification of the example of a non-developable space with a sharp base [2]. We add extra points to a (non-separable) metric space B in such a way that the resulting space is pseudocompact, has a sharp base but is not compact, hence not metrizable.

Let B = ^{ω}𝔠 be the Tychonoff product of countably many copies of the discrete space of size continuum with the usual Baire metric. For each finite partial function f ∈ ^{<ω}𝔠, let [f] denote the basic open subset of B,

[f] = {g ∈ ^{ω}𝔠 : f ⊆ g} (so [f] is the collection of all elements of B which agree with f on dom f). Note that, if dom f ⊆ dom g, then the two basic open sets [f] and [g] have non-empty intersection if and only if f ⊆ g if and only if [g] ⊆ [f]. If [f] ∩ [g] = ∅ then the functions f and g are incompatible (we write f ⊥ g) and neither f ⊆ g nor g ⊆ f.

Let

S = {S ∈ ^{ω}(^{<ω}𝔠) : S(m) ⊥ S(n), for each m and n},

so that each S in S codes for a sequence of disjoint basic open sets in B. Enumerate S as {S_{α} : α ∈ 𝔠} in such a way that each S in S occurs 𝔠 times. To ensure that our space is pseudocompact, we recursively add limit points (to some of) these sequences of open sets. These limit points s_{α} will have basic open neighbourhoods of the form

N(α, n) = {s_{α}} ∪ ⋃_{m≥n}[T_{α}(m)],

where T_{α} ∈ ^{ω}(^{<ω}𝔠) is defined depending on S_{α}.

Suppose that for each α < γ we have either defined if possible a sequence T_{α} ∈ ^{ω}(^{<ω}𝔠) such that

(1γ) for i ≠ j, T_{α}(i) ⊥ T_{α}(j),

(2γ) for β < γ, β ≠ α, T_{β} defined, ran T_{α} ∩ ran T_{β} = ∅, and

(3γ) for β < γ, β ≠ α, T_{β} defined, if T_{α}(i) ⊇ T_{β}(j), then T_{α}(i′) ⊥ T_{β}(j′) for all ⟨i′, j′⟩ ≠ ⟨i, j⟩

or we have not defined T_{α}. We now define T_{γ}.

First note that if S′_{γ}(i) extends S_{γ}(i), then the open set [S′_{γ}(i)] is a subset of [S_{γ}(i)], so any limit of the sequence of open sets {[S′_{γ}(i)] : i ∈ ω} will also be a limit of the sequence {[S_{γ}(i)] : i ∈ ω}.

Since each T_{α}(j) is finite, there is some δ < 𝔠 which is not in ⋃{T_{α}(j) : α < γ, j ∈ ω}. For each i ∈ ω, let S′_{γ}(i) = S_{γ}(i)^{⌢}{δ} extend S_{γ}(i). Then for all i, j ∈ ω and α < γ, S′_{γ}(i) ⊈ T_{α}(j) and T_{α}(j) ⊆ S′(i) only if T_{α}(j) ⊆ S(i). Notice that this implies that [T_{α}(j)] ⊈ [S′_{γ}(i)] and that [S′_{γ}(i)] ⊆ [T_{α}(j)] only if [S_{γ}(i)] ⊆ [T_{α}(j)].

Case 1: Suppose that there exists some α < γ for which T_{α} was defined, such that for infinitely many i ∈ ω there exists some j ∈ ω such that S′_{γ}(i) ⊇ S_{γ}(i) ⊇ T_{α}(j). In this case we do not define T_{γ} (since infinitely many of the basic open sets [T_{α}(j)] contain an open set [S_{γ}(i)] and the limit point s_{α} will deal with the sequence S_{γ}).

Case 2: Now suppose that Case 1 does not hold and that hence

(*) for each α < γ there are at most finitely many i for which S′_{γ}(i) ⊇ T_{α}(j) for some j.

Suppose further that for each i ≤ k, we have chosen natural numbers 0 = r_{0} < r_{1} < ⋯ < r_{k} and defined T_{γ}(i) to be S′_{γ}(r_{i}).

Since each T_{γ}(i) is a finite partial function, there are at most finitely many possible partial functions such that f ⊆ T_{γ}(i) for some i ≤ k. By condition (2γ) there are at most finitely many α < γ with such an f in ran T_{α}. List these α as α(1), … α(m). By (*), for each α(m), there is a j_{m} such that for all i ≥ j, S′_{γ}(i) does not extend any T_{α(m)}(j). Now let r_{k+1} = max j_{m} and T_{γ}(k + 1) = S′_{γ}(r_{k+1}).

We now claim that conditions (1𝔠), (2𝔠) and (3𝔠) hold. Suppose that T_{β} and T_{α} were defined for some β < α < 𝔠. Condition (1𝔠) is obvious since each T_{α} is a subsequence of S′_{α} each term of which extends the corresponding term of S_{α}, and S_{α} is a sequence of pairwise incompatible partial functions. (2𝔠) holds since, if β < α, then the extension S′_{γ}(i) was chosen to ensure that T_{β}(j) ⊉ S′_{α}(i) for any j, so in particular T_{β}(j) ≠ T_{α}(i) and ran T_{β} ∩ ran T_{α}. To see that (3𝔠) holds, note first that S′_{α}(i) was chosen so that S′_{α}(i) ⊈ T_{β}(j) for any j, which implies that T_{α}(i) ⊈ T_{β}(j) for any ⟨i, j⟩. On the other hand, suppose that i is least such that for some j, T_{β}(j) ⊆ T_{α}(i). If k > i, then T_{α}(k) = S′_{α}(r_{k}) and r_{k} was chosen precisely so that S′_{α}(r_{k}) ⊉ T_{β}(l) for any l ∈ ω. Moreover, there can be at most one j such that T_{α}(i) ⊇ T_{β}(j), since by (1𝔠), T_{β}(j) ⊥ T_{β}(l), j ≠ l. This completes the recursion.

Let L = {s_{α} : T_{α} has been defined} be a set of pairwise distinct points disjoint form B and let P = B ∪L. We topologize P by letting B be an open subspace with the usual Baire metric topology and declaring the n^{th} basic open set about the point s_{α} to be the set N(α, n) = {s_{α}} ∪ ⋃_{m≥n}[T_{α}(m)].

If T_{α} = {[T_{α}(n)] : n ∈ ω}, then condition (1𝔠) ensures that each T_{α} is a pairwise disjoint collection, (2𝔠) ensures that each basic open set [f] occurs in at most one T_{α}, and (3𝔠) ensures that if N(α, n) meets N(β, m), then N(α, n) ∩ N(β, m) = [T_{α}(j)] ∩ [T_{β}(k)] for some j ≥ n and k ≥ m.

That P has a sharp base follows exactly as for the example due to Alleche et al. Let B_{B} be a sharp base for B and let

B = B_{B} ∪ {N(α, n) : s_{α} ∈ L and n ∈ ω}.

Suppose x ∈ ⋂_{k∈ω} B_{k} for some (injective) sequence {B_{k} ∈ B : k ∈ ω}. Since B_{B} is a sharp base and s_{α} ∈ N ∈ B if and only if N = (α, n) for some n, the only case that is not obvious is when x ∈ B and B_{k} = N(α_{k}, m_{k}) for all but finitely many k. But in this case condition (3𝔠) implies that, for n ≥ 1, ⋂_{k≤n} B_{k} = ⋂_{k≤n}[T_{α_{k}}(j_{k})]. Moreover (2𝔠) implies that T_{α_{k}}(j_{k}) ≠ T_{α_{k′}} (j_{k′}), so that {⋂_{k≤n} B_{k} : n ∈ ω} contains a strictly decreasing subsequence and is therefore a base at x.

Since the set {s_{α} : α ∈ 𝔠} is infinite, closed discrete, P is not compact. On the other hand, P is pseudocompact (so P is not metrizable). To see this, suppose that φ is a continuous real-valued function on P taking values in [n, ∞) for each n ∈ ω. Since B is dense in P, for each n ∈ ω, there is some x_{n} in B such that φ(x_{n}) > n. By continuity, {x_{n} : n ∈ ω} does not have a limit point in B. Since φ is continuous and B is metrizable, there are basic open sets [f_{n}] for each n ∈ ω such that x_{n} ∈ [f_{n}] ⊆ φ^{−1}(n, ∞) and {[f_{n}] :∈ ω} is a disjoint collection. But in this case f_{n} ⊥ f_{m} when n ≠ m so that {f_{n} : n ∈ ω} = S_{α} for some α ∈ 𝔠. In which case, either s_{α} and T_{α} were defined or s_{α} was not defined and, for some β < α, T_{β}(j) ⊆ S_{α}(n) = f_{n} for infinitely many n. In the second case, each basic open neighbourhood N(β, n) of s_{β} contains infinitely many of the sets [f_{n}]. In the first case, T_{α} was chosen so that T_{α}(i) ⊇ f_{r_{i}} for each i ∈ ω, so that [T_{α}(i)] ⊆ [f_{r_{i}}]. In either case, each neighbourhood of s_{β} or s_{α} contains points which take arbitrarily large values under φ, contradicting continuity.

Now suppose for a contradiction that P × [0, 1] has a sharp base. We shall show that this would imply that P has a σ-point finite base, which is impossible since Uspenskiĭ [18] shows that a pseudocompact space with a σ-point finite base is metrizable.

To this end, let W be a sharp base for P × [0, 1] and let C be a countable sharp base for [0, 1]. For each x in L choose W_{n}^{x} in W, B_{n}^{x} in B (the sharp base for P), and C_{n}^{x} in C such that B_{n}^{x} × C_{n}^{x} ⊆ W_{n}^{x}, {W_{n}^{x} : n ∈ ω} (and hence {B_{n}^{x} × C_{n}^{x} : n ∈ ω}) is a local base at (x, 1/2) and W_{0}^{x} ∩ (L × [0, 1]) ⊆ {x} × [0, 1], which is possible since L is a closed discrete subset of P.

Let

B_{C} = {B ∈ B : for some n ∈ ω and some x ∈ L, B = B_{n}^{x} and C = C_{n}^{x}}.

If B_{C} is not point finite then for some y in P, y ∈ ⋂_{j∈ω} B_{j} for some pairwise distinct B_{j} ∈ B_{C}. By definition, for each j there is some x_{j} ∈ L and n_{j} ∈ ω such that B_{j} = B_{n_{j}}^{x_{j}} and C = C_{n_{j}}^{x_{j}}. But then

{y} × C ⊆ ⋂_{j∈ω}(B_{n_{j}}^{x_{j}} × C_{n_{j}}^{x_{j}}) ⊆ ⋂_{j∈ω} W_{n_{j}}^{x_{j}}.

Since B_{j} ≠ B_{k}, either there is an infinite set J ⊆ ω such that x_{j} ≠ x_{k}, for distinct j, k ∈ J, or there is an infinite set K ⊆ ω such that x_{j} = x_{k} = x but n_{j} ≠ n_{k} for some x ∈ L and distinct j, k ∈ K. In the first case, {W_{n_{j}}^{x_{j}} : j ∈ J} is a pairwise distinct subset of the sharp base W and ⋂_{j∈J} W_{n_{j}}^{x_{j}} contains at most one point. In the second case ⋂_{k∈K}(B_{n_{k}}^{x_{k}} × C_{n_{k}}^{x_{k}}) = (x, 1/2), since {B_{n}^{x} × C_{n}^{x} : n ∈ ω} is a local base at (x, 1/2). In either case, {y} × C contains at most one point, which is not the case, and B_{C} is point finite.

Since {B_{n}^{x} × C_{n}^{x} : n ∈ ω} is a local base at (x, 1/2) and C is countable, B = ⋃_{C∈C} B_{C} is a σ-point finite base for points of L. But P = B ∪ L] and B is a metric space, so P has a σ-point finite base: a contradiction.

By Theorem 4, P does not have a G_{δ}^{∗} diagonal, nor indeed is it submetacompact. □

So when is a pseudocompact space with a sharp base metrizable? As mentioned above, a pseudocompact, CCC regular space with a sharp base is metrizable [4, Theorem 21]. Pseudocompact, Moore spaces are CCC. Moreover, in proving that a pseudocompact Tychonoff space with a regular G_{δ}-diagonal is metrizable, McArthur [14] proves that a pseudocompact space with a G_{δ}^{∗}-diagonal is developable. Hence we have

Theorem 4. A pseudocompact regular space X with a sharp base is metrizable if either of the following hold:

(1) X is developable, or;

(2) X has a G_{δ}^{∗}-diagonal.

A pseudocompact space with a G_{δ}-diagonal is Čech complete [4, Lemma 20], hence Baire, so the following theorem is a strengthening of Theorem 21 of [4]. A space is strongly quasi-complete if there is a map g assigning to each x ∈ X and n ∈ ω an open set g(n, x) containing x such that {x_{n}} clusters at x whenever {x, x_{n}} ⊆ ⋂_{i≤n} g(i, y_{i}). Weakly developable spaces are clearly strongly quasi-complete.

Theorem 5. A regular, locally CCC, locally Baire space with a sharp base is metrizable.

Proof. Let X be a regular, locally CCC, locally Baire space with a sharp base. Since X has a weak development, it is strongly quasi-complete. Hodel [12] shows that every regular, quasi-complete CCC Baire space with either a G_{δ}-diagonal or a point countable separating open cover is separable. Since X has a sharp base, X has a point countable base, a G_{δ}-diagonal and is quasicomplete. Hence X is locally separable. But every locally separable regular space with a point countable base is a disjoint union of clopen subspaces each of which has a countable base (see Theorem 7.2 of [10]). Hence X is metrizable. □

Generalising the fact that a countably compact space with a sharp base is metrizable we have:

Theorem 6. A regular, ω_{1}-compact space with a sharp base is metrizable.

Proof. Since X is ω_{1}-compact, every point-countable open cover of X has a countable subcover (Lemma 7.5, [10]). Since X has a sharp base, it has a point countable base and therefore is Lindelöf. A metacompact space with a sharp base is developable [2] and so a Lindelöf space with a sharp base is metrizable. □

Not surprisingly a monotonically normal space with a sharp base is metrizable (c.f. [6] where it is shown that a GO-space with a sharp base is metrizable).

Theorem 7. For a monotonically normal space X the following are equivalent:

(1) X is metrizable;

(2) X has a sharp base;

(3) X has a weak development;

(4) X is strongly quasi-complete;

(5) X has a base of countable order and a G_{δ}-diagonal.

Proof. Since 1 ⟹ 2 ⟹ 3 ⟹ 4 ⟹ 5 (that 4 implies 5 follows from Theorems 2.2 and 2.3 of [8]), it remains to show that a monotonically normal space with a base of countable order and a G_{δ}-diagonal is metrizable. By the Balogh-Rudin theorem [5], since a stationary set of a regular cardinal does not have a G_{δ}-diagonal, a monotonically normal space with a G_{δ}-diagonal is paracompact. The result then follows since a paracompact space with a base of countable order is metrizable [3]. □

The proof that P ×[0, 1] does not have a sharp base does not quite extend to a proof that if the product of a space X with [0, 1] has a sharp base then X has a σ-point finite base. The converse however is easily seen to be true.

Proposition 8. If a space X has a σ-point finite sharp base then X × [0, 1] has a sharp base.

Proof. Suppose that B = ⋃ B_{n} is a σ-point finite sharp base for X and C = ⋃ C_{n} is a development for [0, 1] such that each C_{n+1} is finite and refines C_{n} (so that C is also a sharp base for [0, 1]). For each n ∈ ω let W_{n} = {B × C : B ∈ B_{n}, C ∈ C_{n}} and let W = ⋃_{n} W_{n}.

Firstly note that W is a base for X × [0, 1]. If (x, r) is in some open set U, choose n and B ∈ B_{m} such that (x, r) ∈ B ×st(r, C_{n}) ⊆ U. Now for some k ≥ max{m, n}, there is B′ ∈ B_{k} x ∈ B′ ⊆ B. But then, since C_{k} refines C_{n}, if r ∈ C ∈ C_{k}, B′ × C ∈ W_{k} and

(x, r) ∈ B′ × C ⊆ B′ × st(r, C_{k}) ⊆ B × st(r, C_{n}) ⊂ U.

Now suppose that (x, r) ∈ B_{j} × C_{j} = W_{j} ∈ W for distinct W_{j}, j ∈ ω. Each W_{n} is a point finite family since both B_{n} and C_{n} are point finite and so both {B_{j}}_{j∈ω} and {C_{j}}_{j∈ω} are infinite. Since B and C are sharp bases, this implies that {⋂_{j≤n} B_{j} × C_{j} : n ∈ ω} is a local base at (x, r) and W is a sharp base as required. □

Ponomarev, see [10], characterized those spaces with a point countable base as precisely the open s-images of metric spaces (a map is an s-map if it has separable fibres). There is a similar characterization for sharp bases.

Theorem 9. A space X has a sharp base if and only if there is a metric space M with a base B and a continuous open mapping f : M → X such that, whenever x ∈ X and {B_{n} ∈ B : n ∈ ω} is a pairwise distinct collection, if f^{−1}(x) ∩ B_{n} ≠ ∅ for each n ∈ ω, then there exists n_{0} such that for each y ∈ X, if f^{−1}(y) ∩ B_{j} ≠ ∅, for each j ≤ n_{0}, then f^{−1}(y) ∩ B_{0} ≠ ∅.

Proof. Suppose that G is a sharp base for the space X. Let

M = {(G_{n}) ∈ G^{ω} : x ∈ ⋂_{n∈ω} G_{n} for some x ∈ X}

be the subspace of the Baire metric space G^{ω}, with metric d((G_{n}), (H_{n})) = 1/2^{k} where k is least such that G_{n} ≠ H_{n}. Let f : M → X be defined letting f((G_{n})) be the unique element of ⋂_{n∈ω} G_{n} and let B be the base for M consisting of all 1/2^{n}-balls about points of M. Then f is easily seen to be a continuous, open mapping onto X and the condition on B in the statement of the theorem is merely a translation of the fact that G is a sharp base. □

It is clear from the proof that, in the statement of the theorem, we can take B to be the collection of 1/2^{n} balls for any n rather than a base for M. Since a space with a sharp base has a point countable sharp base, we can also assume that the map in the statement of the theorem is an s-map. However, it is not immediately clear that we can prove that a space with a sharp base has a point countable base directly from the theorem.

We conclude with some open problems. Since every collectionwise normal Moore space is metrizable, the following is a natural and intriguing question.

Question 1. Is every collectionwise normal space with a sharp base metrizable?

Example 4 of [2] shows that weakly developable, collectionwise normal spaces do not have to be metrizable and the Heath V-space over a Q-set is an example of a normal space with a uniform base that is not metrizable. On the other hand, the answer is ‘yes’ if the space is also submetacompact (since it is then a Moore space) or a strict p-space. We might also ask whether a perfect, collectionwise normal space with a sharp base is metrizable. It is interesting to note that it is not known whether a collectionwise normal space with a point countable base need be paracompact.

Since the Heath V-space over a Δ-set is countably paracompact but not normal [13], at least consistently a countably paracompact, (Moore) space with a sharp base need not be normal. What about the converse?

Question 2. Is there a Dowker space with a sharp base?

Question 3. Is every perfectly regular space with a sharp base developable? Is every normal space with a sharp base developable? Is every perfectly regular, pseudocompact space with a sharp base metrizable?

Not every Moore space with a weakly uniform base has a uniform base (see [2]) so we ask:

Question 4. Does every Moore space with a sharp base have a uniform base?

Every pseudocompact space with a G_{δ}-diagonal is Čech complete [4], and every pseudocompact Moore space with a sharp base is metrizable.

Question 5. Is every Čech complete Moore space with a sharp base metrizable? What about Baire instead of Čech complete?

Question 6. If X × [0, 1] has a sharp base, does X have a σ-point finite sharp base?

Question 7. Does the image (or pre-image) of a space with a sharp base under a perfect map (closed and open map, open map with compact, countable or finite fibres) have a sharp base?

Acknowledgements. Much of this research took place whilst the first author held a Universitas 21 Traveling Fellowship at the Department of Mathematics at the University of Auckland. He would like to thank the department for their hospitality and both the department and U21 for their financial support.

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School of Mathematics and Statistics, University of Birmingham, Birmingham, B15 2TT, UK

E-mail address: c.good@bham.ac.uk

Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford OX1 3LB, UK

E-mail address: knight@maths.ox.ac.uk

Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand

E-mail address: mohamad@math.auckland.ac.nz