����������� ������ 46 (1994), 81–87

UDC 515.122.2

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PRE-LINDELÖF QUASI-PSEUDO-METRIC AND QUASI-UNIFORM SPACES

H.-P. A. Künzi, M. Mršević, I. L. Reilly, M. K. Vamanamurthy

Abstract. When the finiteness condition in the definition of precompactness is replaced by countability, a new notion in the class of quasi-uniform spaces is introduced, and we call it pre-Lindelöfness. Some properties of pre-Lindelöf spaces, in particular in the subclass of quasipseudo-metric spaces, are investigated.

1. Introduction

Recently, extensive studies of precompact quasi-uniform spaces have been undertaken. Precompactness is a weaker form of generalised total boundedness. Let us recall that a quasi-uniform space (X, U) is called totally bounded if for each U ∈ U there is a finite cover { A_{i} : i = 1,…, n } of X such that A_{i} × A_{i} ⊂ U for each i = 1,…, n. Furthermore (X, U) is called precompact if for each U ∈ U there is a finite subset F ⊂ X such that X = ⋃_{x∈F} U(x) = U(F). As usual U(x) = { y ∈ X : (x, y) ∈ U }.

If the finiteness condition in the definition of precompactness is replaced by countability, a new property of quasi-uniform spaces is introduced, and we call it pre-Lindelöfness.

Definition 1. Let (X, U) be a quasi-uniform space. Then (X, U) is called pre-Lindelöf if for each U ∈ U there is a countable subset C ⊂ X such that X = ⋃_{c∈C} U(c) = U(C). A space X is hereditarily pre-Lindelöf if each subspace of X is pre-Lindelöf.

In the class of uniform spaces the pre-Lindelöf property is called by different names, one of which is trans-separability. ( See for example [8].) This property is strictly weaker than both Lindelöfness and separability, and has proved useful in investigations of metrisability of topological vector spaces.

Keywords and phrases: quasi-pseudo-metric, quasi-uniform, (hereditarily) Lindelöf, (hereditarily) precompact, (hereditarily) pre-Lindelöf, separable.

AMS(1991) Subject classification: 54E15, 54E99, 54D20

This paper was written while the first author was supported by the Spanish Ministry of Education and Sciences under the DGICYT grant SAB94-0120 and the second author was supported by Science Fund of Serbia under grant 0401A.

All (hereditarily) Lindelöf spaces and all (hereditarily) precompact spaces are (hereditarily) pre-Lindelöf, but the converse is not true in general, except for hereditary Lindelöfness in the class of quasi-pseudo-metric spaces as Theorem 2 shows.

As noted in [7], the usual metric on the set of real numbers gives a space which is hereditarily Lindelöf but not precompact, while the Sorgenfrey plane is pre-Lindelöf but not Lindelöf. The Sorgenfrey line quasi-metric restricted to the interval [0, 1] is precompact. Its square is precompact but not Lindelöf.

In the class of quasi-pseudo-metric spaces pre-Lindelöfness has some nice properties. We prove (Theorem 1) that pre-Lindelöfness is equivalent to separability of the conjugate space. Also, we show that hereditary pre-Lindelöfness is equivalent to the hereditary Lindelöf property (Theorem 2). Neither of these two statements is true in general in the class of quasi-uniform spaces.

Our notation is standard. In particular, R and N denote the sets of real numbers and positive integers, respectively. In a quasi-pseudo-metric space (X, d) the ball with centre x and radius ε is the set { y ∈ X | d(x, y) < ε } and is denoted by B_{d}(x, ε), although we suppress the ’d’ when there is no possible confusion. If d is a quasi-pseudo-metric on X, then so is d′ defined by d′(x, y) = d(y, x) for x, y ∈ X, and d′ is called the conjugate of d.

2. Pre-Lindelöf quasi-pseudo-metric spaces.

In the subclass of quasi-pseudo-metric spaces pre-Lindelöfness can be defined in the following way.

Definition 2. A quasi-pseudo-metric space (X, d) is pre-Lindelöf if for every ε > 0 there is a countable subset C ⊂ X such that X ⊂ ⋃_{c∈C} B(c, ε).

The duality between pre-Lindelöfness and separability is given by the following result.

Theorem 1. A quasi-pseudo-metric space (X, d) is pre-Lindelöf if and only if its conjugate space (X, d′) is separable.

Proof. Let (X, d) be pre-Lindelöf. For each n ∈ N there is a countable subset A_{n} ⊂ X such that X = ⋃_{a∈A_{n}} B_{d}(a, \frac{1}{n}). The set A = ⋃_{n∈N} A_{n} is countable and dense in (X, d′).

Conversely, let D be a countable dense subset in (X, d′). Let ε > 0 and let x ∈ X. There is an a ∈ D such that d′(x, a) < ε, which implies d(a, x) < ε. Thus X = ⋃_{a∈D} B_{d}(a, ε), so that (X, d) is a pre-Lindelöf space.

Corollary 1. In a pseudometric space the following are equivalent:

(i) second countability,

(ii) separability, (iii) Lindelöfness, (iv) ccc ( the countable chain condition),

(v) pre-Lindelöfness.

Corollary 2. If Y is an open subset in the conjugate topology of a quasipseudo-metric space (X, d_{X}), then (X, d_{X}) is pre-Lindelöf implies (Y, d_{Y}) is pre-Lindelöf.

While pre-Lindelöfness is in general strictly weaker than the Lindelöf property, their hereditary counterparts coincide in the class of quasi-pseudo-metric spaces, as the following result shows.

Theorem 2. A quasi-pseudo-metric space is hereditarily pre-Lindelöf if and only if it is hereditarily Lindelöf.

Proof. Only the ”only if” part has to be proved. For this we modify the proof of Lemma 2 in [7].

Let E ⊂ X and let G be a collection of open sets in X such that E ⊂ ⋃{ G | G ∈ G }. For each x ∈ E choose G(x) ∈ G and n(x) ∈ N such that B(x, 1/n(x)) ⊂ G(x). For each n ∈ N let A_{n} = { x ∈ E | n(x) = n }. For each n ∈ N there is countable subset C_{n} of A_{n} such that A_{n} ⊂ ⋃_{c∈C_{n}} B(c, \frac{1}{n}). Then E ⊂ ⋃\{ B(c, \frac{1}{n}) | c ∈ C_{n} and n ∈ N }. It follows that E is a Lindelöf subset of X.

Since both hereditary Lindelöfness and separability imply the countable chain condition, we have the following

Corollary 3. Let (X, d) be a hereditarily (pre-)Lindelöf quasi-pseudo-metric space. Then both (X, d) and (X, d′) are hereditarily ccc.

Note that Example 5 in [7] gives a space (X, s) which is hereditarily precompact, thus hereditarily (pre-)Lindelöf, but not separable. Thus hereditary (pre-)Lindelöfness and separability are not comparable; also, neither separability nor hereditary (pre-)Lindelöfness is equivalent to the ccc in quasi-pseudo-metric spaces.

Corollary 4. [4, Theorem 4] A quasi-pseudo-metric space is hereditarily (pre-)Lindelöf if and only if its conjugate space is hereditarily separable.

We are able to define a more general version of the pre-Lindelöf property given in Definition 2.

Definition 3. Let (X, d) be a quasi-pseudo-metric space. Given D ⊂ X and ε > 0, we define (X, d) to be (D, ε)-Lindelöf if X = _{⋃x∈D}B_{d}(x, ε).

Definition 4. Let (X, d) be a quasi-pseudo-metric space. Given D ⊂ X and ε > 0, we define D to be ε-dense in (X, d), if for every x ∈ X there is a point y ∈ D ∩ B_{d}(x, ε).

Remark. (X, d) is pre-Lindelöf (pre-compact) if for every ε > 0 there is a countable (finite) subset D such that (X, d) is (D, ε)-Lindelöf.

Theorem 3. (X, d) is separable if and only if for every ε > 0 there is a countable subset D which is ε-dense in (X, d).

Proof. The ”only if” part is obvious. For the converse, for every n ∈ N there is a countable subset D_{n} such that D_{n} is \frac{1}{n}-dense in (X, d). Let D = _{⋃n∈N}D_{n}. Then D is countable. We claim that D is dense in (X, d). Let x ∈ X and ε > 0 be given. Choose n ∈ N, with \frac{1}{n} < ε. Then there is a point y ∈ D_{n}∩B_{d}(x, \frac{1}{n}) ⊂ D∩B_{d}(x, ε).

Theorem 4. Let d, d′ be conjugate quasi-pseudo-metrics on X and let D ⊂ X, ε > 0. Then (X, d) is (D, ε)-Lindelöf if and only if D is ε-dense in (X, d′).

Proof. Let x ∈ X, y ∈ D. Then x ∈ B_{d}(y, ε) if and only if y ∈ B_{d′}(x, ε).

3. Pre-Lindelöf quasi-uniform spaces.

When the class of quasi-pseudo-metric spaces is enlarged to the class of quasiuniform spaces, pre-Lindelöfness behaves in a slightly different way. Theorems 1 and 2 no longer hold in general as the following examples show.

Example 1. Let X be an uncountable set endowed with the cocountable topology T . Let U be the Pervin quasi-uniformity compatible with T . The space (X, U) is totally bounded, thus both (X, U) and its conjugate (X, U^{−1}) are hereditarily precompact, hence hereditarily pre-Lindelöf. The topology T (U^{−1}) is discrete, thus the space (X, U^{−1}) is neither separable nor Lindelöf. Furthermore (X, T ) = (X, T (U)) is not separable. Note that (X, U) has the ccc while (X, U^{−1}) has not.

Example 2. For any topological space X the semi-continuous quasi-uniformity S(X) has the property that S(X)^{∗} = sup{S(X), S(X)^{−1}} is (hereditarily) pre-Lindelöf. (See [2, Theorem 2.12] and Proposition 2.) Thus Theorem 2 does not hold even in the class of uniform spaces.

Precompactness is a productive property as stated in [2] for quasi-uniform spaces. Theorem 5 shows that pre-Lindelöfness is also productive. In Theorem 5 of [7] it was proved that hereditary precompactness is productive as well, while the Sorgenfrey plane shows a different behaviour of hereditary pre-Lindelöfness. By Theorem 2, the Sorgenfrey plane being quasi-metrisable is not hereditarily pre-Lindelöf, thus hereditary pre-Lindelöfness is not a productive property. Theorem 6 gives a necessary and sufficient condition for a product of infinitely many quasiuniform factor spaces to be hereditarily pre-Lindelöf.

The next two results are given without proofs which are straightforward.

Proposition 1. Let (X, U) and (Y, V) be quasi-uniform spaces and let f: (X, U) → (Y, V) be a quasi-uniformly continuous function. If (X, U) is (hereditarily) pre-Lindelöf, then so is (f(X), V_{f(X)}).

Proposition 2. Pre-Lindelöfness is hereditary in uniform spaces.

Theorem 5. The nonempty product of a family of quasi-uniform spaces is pre-Lindelöf if and only if each factor space is pre-Lindelöf.

Proof. The ”only if” part follows from Proposition 1. For the converse, let (X, U) be the product of a family {(X_{i}, U_{i})}_{i∈I} of pre-Lindelöf quasi-uniform spaces. For each j ∈ I let pr_{j} : ∏_{i∈I} X_{i} → X_{j} denote the projection. Suppose that U = ⋂_{k=1}^{n}(pr_{i_{k}} × pr_{i_{k}})^{−1}(V_{i_{k}}), where V_{i_{k}} ∈ U_{i_{k}}, k = 1,…, n, and n ∈ N. For each V_{i_{k}} there is a countable subset C_{i_{k}} ⊂ X_{i_{k}} such that X_{i_{k}} = ⋃_{c∈C_{i_{k}}} V_{i_{k}}(c). For each i ∈ I \ {i_{1},…, i_{n}} choose a point a_{i} ∈ X_{i}. Let C be a subset of X consisting of all points c ∈ X such that c_{i} ∈ C_{i} if i ∈ {i_{1},…, i_{n}}, and c_{i} = a_{i} otherwise. The subset C is countable and X = U(C). Therefore ∏_{i∈I} X_{i} is pre-Lindelöf.

Theorem 6. The nonempty product of a family {(X_{i}, U_{i})}_{i∈I} of hereditarily pre-Lindelöf quasi-uniform spaces is hereditarily pre-Lindelöf if and only if the product ∏_{i∈F} X_{i} is hereditarily pre-Lindelöf for any nonempty finite subset F of I.

Proof. Since hereditary pre-Lindelöfness is preserved by quasi-uniformly continuous surjections, the ”only if” part is clear. For the converse, let A be a subspace of ∏_{i∈I} X_{i}. For each j ∈ I let pr_{j}: ∏_{i∈I} X_{i} → X_{j} denote the projection. Suppose that U = ⋂_{k=1}^{n}(pr_{i_{k}} × pr_{i_{k}})^{−1}(V_{i_{k}}), where V_{i_{k}} ∈ U_{i_{k}}, k = 1,…, n, and n ∈ N. Let p : ∏_{i∈I} X_{i} → ∏_{k=1}^{n} X_{i_{k}} be the projection. For simplicity the projections of the latter product onto its factor spaces are also denoted by pr_{i}. Since ∏_{k=1}^{n} X_{i_{k}} is hereditarily pre-Lindelöf, there is a countable subset C ⊂ p(A) such that p(A) ⊂ ⋃_{c∈C}[V_{i_{1}}(pr_{i_{1}}(c)) × ⋯ × V_{i_{n}}(pr_{i_{n}}(c))]. For each c ∈ C choose d_{c} ∈ A such that p(d_{c}) = c. Therefore A ⊂ ⋃_{c∈C} U(d_{c}) and we conclude that ∏_{i∈I} X_{i} is hereditarily pre-Lindelöf.

Proposition 3. Let (X, U) be a quasi-uniform space such that sup{U, U^{−1}} is (hereditarily) pre-Lindelöf and let (Y, V) be a hereditarily pre-Lindelöf quasiuniform space. Then (X × Y, U × V) is hereditarily pre-Lindelöf.

Proof. Let B ⊂ X × Y and let U ∈ U and V ∈ V. Since sup{U, U^{−1}} is pre-Lindelöf, there exists a countable cover (A_{i})_{i∈N} of { x ∈ X | (x, y) ∈ B } such that A_{i} × A_{i} ⊂ U whenever i ∈ N. For each i ∈ N set D_{i} = { y ∈ Y | there is x ∈ A_{i} with (x, y) ∈ B }. Since (Y, V) is hereditarily pre-Lindelöf, for each i ∈ N there is a countable set C_{i} ⊂ D_{i} such that D_{i} ⊂ V (C_{i}). For each i ∈ N and each c ∈ C_{i} choose x_{c} ∈ A_{i} such that (x_{c}, c) ∈ B. Set C = { (x_{c}, c) | c ∈ C_{i}, i ∈ N }. Let (x, y) ∈ B. There is i_{0} ∈ N such that x ∈ A_{i_{0}}. Therefore y ∈ D_{i_{0}} and y ∈ V (c) for some c ∈ C_{i_{0}}. Thus (x, y) ∈ A_{i_{0}} × V (c) ⊂ U(x_{c}) × V (c). We conclude that (X × Y, U × V) is hereditarily pre-Lindelöf.

Theorem 7. The product (X ×Y, U ×V) of a hereditarily pre-Lindelöf quasiuniform space (X, U) and a hereditarily precompact quasi-uniform space (Y, V) is hereditarily pre-Lindelöf.

Proof. Suppose that a subspace B of X × Y is not pre-Lindelöf. Then there are U ∈ U and V ∈ V such that B ⊂ ⋃_{a∈A} W (a) and A ⊂ B imply that A is uncountable, where W = [(pr_{X} × pr_{X})^{−1}(U)] ∩ [(pr_{Y} × pr_{Y})^{−1}(V)] and pr_{X}: X × Y → X and pr_{Y} : X × Y → Y denote the projections. Inductively construct C = { x_{α} | α < ω_{1} } such that x_{α} ∈ B \ W ({ x_{β} | β < α }) whenever α < ω_{1}. Define a graph with vertex set C and edges {x_{α}, x_{β}} provided that α < β < ω_{1} and pr_{X}(x_{β}) ∈ U(pr_{X}(x_{α})). Since (X, U) is hereditarily pre-Lindelöf, for each uncountable D ⊂ C there are two vertices of D connected by an edge. By [1, Theorem 5.22], there is a countably infinite subset E of C such that any two vertices of E are connected. Hence if α < β < ω_{1} and x_{α}, x_{β} ∈ E, then pr_{Y} (x_{β}) ∉ V (pr_{Y} (x_{α})). We conclude that the first infinite initial segment of pr_{Y} (E) ordered according to the index set is not precompact, a contradiction. Therefore X × Y is hereditarily pre-Lindelöf.

Theorem 8. A topological space X is hereditarily Lindelöf if and only if each of its compatible quasi-uniformities is hereditarily pre-Lindelöf.

Proof. Only the ”if” part needs a proof. If X is not hereditarily Lindelöf, then there is a strictly increasing sequence (G_{α})_{α<ω_{1}} of open sets in X . The binary relation T defined by T (x) = ⋂{ G_{α} | x ∈ G_{α}, α < ω_{1} } for all x ∈ X, where we use the convention that the intersection over the empty collection is X, belongs to the fine quasi-uniformity V of X. Since V restricted to the subspace ⋃_{α<ω_{1}} G_{α} is not pre-Lindelöf, V is not hereditarily pre-Lindelöf.

Problem. Characterize those topological spaces that admit only pre-Lindelöf quasi-uniformities.

Proposition 4. If (X, U) and (Y, V) are hereditarily pre-Lindelöf, then each subspace A of X × Y has the property that for each W ∈ U × V there is M ⊂ A such that |M| ≤ 2^{ℵ_{0}} and A ⊂ W (M).

Proof. Otherwise there are U ∈ U, V ∈ V and C = { x_{α} | α < (2^{ℵ_{0}})^{+} } such that x_{α} ∉ U(pr_{X}(x_{β})) × V (pr_{Y} (x_{β})) whenever β < α < (2^{ℵ_{0}})^{+} . Let β < α < (2^{ℵ_{0}})^{+}. If pr_{X}(x_{α}) ∉ U(pr_{X}(x_{β})), set f({x_{β}, x_{α}}) = 1, and set f({x_{β}, x_{α}}) = 0 otherwise. Since X and Y are hereditarily pre-Lindelöf we see similarly as above that there is no uncountable subset A of C such that f is constant on [A]^{2}, contradicting the well known Erdös-Rado Partition Theorem [3, Theorem 69].

Recall that a topological space is almost realcompact if each maximal open filter F such that \overline{F} = \{ \overline{F} | F ∈ F \} has the countable intersection property has a cluster point. The space is called closed complete if each maximal closed filter with the countable intersection property has a non-empty intersection. (See [6, p. 360].) A quasi-uniform space (X, U) is called complete if each U-Cauchy filter has a cluster point. (See [2, Definition 3.8].)

Proposition 5. Each pre-Lindelöf complete quasi-uniform space (X, U) is:

(i) almost realcompact, (ii) closed complete.

Proof. (i) Let F be a maximal open filter in (X, U) such that \overline{F} has the countable intersection property. Consider U ∈ U. There is a V ∈ U such that V (x) is open in (X, T (U)) whenever x ∈ X [2, p. 3] and such that V ⊂ U. Since U is pre-Lindelöf, there is a countable subset C of X such that ⋃_{x∈C} V (x) = X. Since \overline{F} has the countable intersection property, there is an x ∈ C such that V (x)∩F ≠ ∅ whenever F ∈ F. By maximality of F, V (x) ∈ F. We conclude that F is a U-Cauchy filter base. Hence ⋂ \overline{F} ≠ ∅ because (X, U) is complete. We have shown that (X, T (U)) is almost realcompact.

(ii) Closed completeness is proved in a similar way.

An analogue of the well known result in the class of topological spaces that compactness is equivalent to countable compactness and Lindelöfness, is the following, whose straightforward proof is omitted. The statement for quasi-pseudo-metric spaces then follows from Corollary 2 of [5].

Proposition 6. A countably compact pre-Lindelöf quasi-uniform space is precompact. In particular, a countably compact pre-Lindelöf quasi-pseudo-metric space is compact.

That the converse of Proposition 6 does not hold can be seen from Example 3. Example 4 shows that the statement is not true for hereditary counterparts even in quasi-metric spaces.

Example 3. The subspace (0, 1) of the real line is precompact but not countably compact.

Example 4. Let X = N and for m, n ∈ X let

d(m, n) = \{ \begin{matrix}\frac{1}{n}, & if m = 1 and n > 1, \\ 0, & if m = n , \\ 1, & otherwise . \end{matrix}

The space (X, d) is compact but not hereditarily precompact.

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(received 15.06.1994.)

H.-P.A. Künzi, Department of Mathematics, University of Berne, 3012 Berne, Switzerland. M. Mršević, Faculty of Mathematics, University of Belgrade, Belgrade, Yugoslavia.

I.L. Reilly and M.K. Vamanamurthy, Department of Mathematics, University of Auckland, Auckland, New Zealand.