@
Appl. Gen. Topol. 18, no. 1 (2017), 23-30

doi:10.4995/agt.2017.4092

c© AGT, UPV, 2017

A note on unibasic spaces and transitive
quasi-proximities

Adalberto Garćıa-Máynez a and Adolfo Pimienta Acosta b,∗

a Instituto de Matemáticas, Universidad Nacional Autónoma de México; Area de la Investigación

Cient́ıfica Circuito Exterior, Ciudad Universitaria Coyoacán, 04510. México, D. F.

(agmaynez@matem.unam.mx)
b Departamento de Matemáticas,Universidad Autónoma Metropolitana, prolongación canal de

miramontes #3855, Col. Ex-Hacienda San Juan de Dios, Delegación Tlalpan, C.P. 14387, México,

D.F-México (pimienta@xanum.uam.mx)

Communicated by M. Sanchis

Abstract

In this paper we prove there is a bijection between the set of all annular
bases of a topological spaces (X,τ) and the set of all transitive quasi-
proximities on X inducing τ. We establish some properties of those
topological spaces (X,τ) which imply that τ is the only annular basis

2010 MSC: Primary 54E05; 54E15; 54D35; Secondary 05C50.

Keywords: annular basis; entourage; semi-block; quasi-proximity; transitive
quasi-proximity-uniformity; unibasic spaces.

1. Introduction

W. J. Pervin showed in [9] that every topological spaces (X,τ) has a quasi-
proximity δ which induces the original topology. In this paper we give condi-
tions for a topological space (X,τ) admits a unique compatible quasi-proximity
in which the topology is the only annular basis.

∗The second author was supported in part by Universidad de la Costa(CUC-
www.cuc.edu.co) under grant of Department of Exact and Natural Sciences, address street
58 # 55-66, Barranquilla, Colombia.

Received 27 August 2015 – Accepted 06 November 2016

http://dx.doi.org/10.4995/agt.2017.4092


A. Garćıa-Máynez and A. Pimienta Acosta

By a quasi-proximity (see [1]) on a set X we will mean a relation δ between
the family of subsets of X satisfying the following axioms:

a) (X,∅) /∈ δ and (∅,X) /∈ δ;
b) (C,A∪B) ∈ δ if only if (C,A) ∈ δ or (C,B) ∈ δ;
c) (A∪B,C) ∈ δ if only if (A,C) ∈ δ or (B,C) ∈ δ;
d) For every x ∈ X, ({x},{x}) ∈ δ;
e) If (A,B) /∈ δ, there exists a set C ⊆ X such that (A,C) /∈ δ and

(X \C,B) /∈ δ.
A quasi-proximity δ on X is a proximity on X if δ = δ−1, i.e., (A,B) ∈ δ iff
(B,A) ∈ δ.

For brevity, we write AδB instead of (A,B) ∈ δ and AδB instead of (A,B) 6∈
δ.

Let δ be a quasi-proximity on a set X. For each A ⊆ X, define à = {x ∈
X : {x}δA}. Then the assignment A → Ã is a Kuratowski-closure operator on
X and the corresponding topology on X is denoted as τδ (see [1]) , 1.27).

H.-P. Künzi and M. J. Pérez-Peñalver in [6] prove some interesting results
about the number of quasi-proximities that a topological spaces admits. H.-P.
Künzi in [3] studies the number of quasi-uniformities belonging to the Pervin
quasi-proximity class.

J. Ferrer in [2] trying to solve the question of whether every T1 topological
space with a unique compatible quasi-proximity should be hereditarily compact,
he shows that it is true for product spaces as well as for locally hereditarily
Lindelöf spaces.

H.-P. Künzi and S. Watson in [7] construct a T1-space X is not hereditarily
compact, but each open susbset of X is the intersection of two compact open
sets. The construction is carried out in ZFC, but the cardinality of the space
is very large.

2. Unibasic spaces and transitive quasi-proximities

The main result of this section establishes a bijection between all annular
bases of a topological space (X,τ) and all transitive quasi-proximities on X
inducing τ.

A basis B for a topological space (X,τ) is annular if it satisfies the following
conditions:

i) ∅ ∈B y X ∈B;
ii) B1,B2 ∈B implies that B1 ∩B2 ∈B and B1 ∪B2 ∈B.

Definition 2.1.

(1) An open set V in (X,τ) is everywhere basic (e.b.) if V belongs to every
annular basis of X.

(2) A topological space (X,τ) is unibasic if τ is the only annular basis of
X.

(3) (X,τ) is minimally basic if X has annular basis B0 which is contained
in every other annular basis B of X.

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A note on unibasic spaces and transitive quasi-proximities

Remark 2.2.

i) Every element of a minimum annular basis B0 of X is e.b. and every
unibasic space is minimally basic.

ii) Every open and compact subset of a topological space X is e.b.. Hence,
every hereditarily compact space is unibasic.

Lemma 2.3. Let B be an annular basis of a topological space (X,τ). Define
AδB iff A ∩ H 6= ∅ for every H ∈ C(B) which contains B. Then δ is a
transitive quasi-proximity on X which induces τ.

Proof. Clearly Xδ∅ and ∅δX. If (A ∪ B)δC, we must have AδC or BδC.
Indeed, AδC and BδC imply the existence of H1,H2 ∈ C(B) such that H1 ∩
H2 ⊇ C, A∩H1 = ∅ = B∩H2. Therefore (A∪B)∩H1 ∩H2 = ∅ and H1 ∩H2
is an element of C(B) containing C, that, (A ∪ B)δC, a contradiction. In a
similar way one may prove that Cδ(A ∪ B) implies that CδA or CδB. It is
obvious that {x}δ{x} for each x ∈ X. Finally, suppose that AδB. Therefore,
there exists an element H ∈ C(B) such that H ⊇ B and A∩H = ∅. Therefore,
AδH and (X \H)δB.

Observe now that (X\H)δH for every H ∈ C(B) and T(X\H,H) = X×X\
[(X\H)×H] = (H×X)∪[X×(X\H)]. Hence, if AδB and H ∈ C(B) satisfies
B ⊆ H ⊆ X\A, we have T(X\H,H) ⊆ [(X\A)×X]∪[X×(X\B)] = T(A,B).
This proves that the quasi-uniformity Uδ is transitive.

Finally, we must prove that τδ = τ. For this, take any set C ⊆ X and
consider the set C1 = {x ∈ X : {x}δC}. It is enough to prove that C1 = C.
If x ∈ X \ C, there exists a set B ∈ B such that x ∈ B ⊆ X \ C. Therefore,
X \B ∈ C(B) and X \B ⊇ C, that is, {x}δC and X \C ⊆ X \C1. On the
other hand, if x ∈ X \C1, i.e., if {x}δC, there exists a set H ∈ C(B) such that
H ⊇ C y x 6∈ H. Therefore, x ∈ X \C and the proof is complete. �

A quasi-proximity δ on a set X is:

(1) Point-symmetric if Aδ{x} implies {x}δA. Equivalently, δ is point-
symmetric if τδ ⊆ τδ−1 .

(2) Locally-symmetric if AδG for every τ-neighborhood G of x implies that
{x}δA.

Notation 2.4. If G is a family of subsets of X, we define: C(G) = {H : X\H ∈
G}.

Let B be an annular basis of a topological space (X,τ) is:
i) Disjunctive (or a Wallman basis ) if whenever x ∈ B ∈ B, there exists

an element Hx ∈ C(B) such that x ∈ Hx ⊆ B.
ii) Regular if whenever x ∈ B ∈B, there exists an element D ∈B and an

element H ∈ C(B) such that x ∈ D ⊆ H ⊆ B.
iii) Normal is for every pair H,K of disjoint elements of C(B), there exists

a pair B,D of disjoint elements of B such that H ⊆ B and K ⊆ D.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 25



A. Garćıa-Máynez and A. Pimienta Acosta

Theorem 2.5. Let B be an annular basis of a topological space (X,τ) and let
δ be the quasi-proximity on X associated to B. Then:

i) B is disjunctive iff δ is point-symmetric.
ii) B is regular iff δ is locallly symmetric.

iii) B is normal iff δ is of Wallman type1.

Proof. We prove only iii). Suppose δ is of Wallman type and let H,K ∈ C(B)
be disjoint. Since H and K are δ- remote, there exists a neighborhood G of H
such that Hδ(X\G) and KδG. This last condition implies the existence of an
elements of H1 ∈ C(B) such that K ⊆ X \ H1 ⊆ X \ G. The first condition
implies the existence of an element K1 ∈ C(B) such that X\G ⊆ K1 ⊆ X\H.
Hence, X \K1 and X \H1 are disjoint elements of B and B is normal.

Assume now that B is normal. Let A,B be δ-remote. Let H,K ∈ C(B)
be disjoint sets such that A ⊆ H and B ⊆ K. Since B is normal, there exist
disjoint elements C,D ∈B such that H ⊆ C and K ⊆ D. Defining G = C, we
have Hδ(X \G) and KδG, i.e., δ is of Wallman type. �

Corollary 2.6. Every transitive point-symmetric quasi-proximity of Wallman
type is locally symmetric and its induced topology is completely regular.

Lemma 2.7. Let δ be a transitive quasi-proximity on a topological space (X,τ)
and suppose that τδ = τ. Then B = {V ∈ τ : V δ(X \V )} is an annular basis
of (X,τ).

Proof. Clearly ∅ ∈B and X ∈B. Suppose now that B1,B2 both belong to B.
If B1∪B2 6∈ B, we would have (B1∪B2)δ(X\B1)∩(X\B2). Therefore B1δ(X\
B1)∩(X\B2) or B2δ(X\B1)∩(X\B2). This would imply that B1δ(X\B1)
or B2δ(X \B2), a contradicition. Hence, B1 ∪B2 ∈B. In a similar fashion we
prove that B1∩B2 ∈B. It remains to prove that B is a basis of (X,τ). Suppose
then that x ∈ V ∈ τ. Therefore {x}δ(X \V ) (recall τδ = τ). Let R ∈Uδ be a
transiive entourage contained in T({x},X \V ). Let us prove that R(x) ⊆ V .
If y ∈ R(x), we have (x,y) ∈ R ⊆ T({x},X\V ) = [(X\{x})×X]∪ [{x}×V ].
Therefore, (x,y) ∈{x}×V , that is, y ∈ V . Besides, R(x)δ(X \R(x)) because
R(x)δ(X\R(x)) would imply that [R(x)×(X\R(x))]∩S 6= ∅ for every S ∈Uδ,
and, in particular, [R(x)×(X \R(x))]∩R 6= ∅. But since R is transitive, this
last statement is clearly false. Hence, we must have that R(x)δ(X \ R(x)).
Since this implies that R(x) ∩ (X \R(x)) = ∅, we deduce that R(x) is open.
Therefore, R(x) ∈B and B is an annular basis of (X,τ). �

Let (X,τ) be a topological space with topology τ. for G ∈ τ let SG =
(G × G) ∪ ((X \ G) × X). The filter generated by {SG : G ∈ τ} is a quasi-
uniformity P for X called Pervin quasi-uniformity (see [8]).

1Two sets A,B ⊆ X are said to be δ-remote if there exist disjoint sets H,K ⊆ X such
that A ⊆ H, B ⊆ K, (X \ H)δH and (X \ K)δK. A quasi-proximity δ on a set X is of
Wallman type if for every pair of δ-remote sets A,B, there exists a neighborhood G of A
such that Aδ(X \G) and BδG.

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A note on unibasic spaces and transitive quasi-proximities

Theorem 2.8. Let (X,τ) be a topological space.Then there exists a bijective
correspondence between the collection of annular bases of (X,τ) and the col-
lection of totally bounded transitive quasi-proximities on X which induce τ.
Hence, (X,τ) is minimally basic iff the family of totally bounded transitive
quasi-uniformities on X inducing τ has a minimum element and (X,τ) is uni-
basic iff P = Uδ0 is the only totally bounded transitive quasi-uniformity inducing
τ.

Theorem 2.9. Let B be an everywhere-basic set on a topological space (X,τ)
and suppose that B 6= X. If K ⊆ X is closed and K ⊆ B, then K is compact.

Proof. Suppose that K is not compact. Then there exists a family G = {Bi : i ∈
J}⊆ τ such that K ⊆∪{Bi : i ∈ J}⊆ B, but for each finite subset J0 ⊆ J, we
have K\∪{Bi : i ∈ J0} 6= ∅. If B′ = {L ∈ τ : L ⊆∪{Bi : i ∈ J0} for some J0 ⊆
J, finite} and let B′′ = {L ∈ τ : L ∩ K = ∅} ∪ {X}, it is easy to check
that B = {L1 ∪ L2 : L1 ∈ B′ and L2 ∈ B′′} is an annular basis of (X,τ).But
B 6∈ B, contradicting the fact that B is everywhere basic. Hence, K must be
compact. �

Definition 2.10. A topological space (X,τ) is R0 if whenever x ∈ V ∈ τ
there exists a closed set Hx such that x ∈ Hx ⊆ V and (X,τ) is R1 if whenever
x,y ∈ X and {x} 6= {y}, there exist disjoint open sets V,W such that x ∈ V
and y ∈ W .

A topological space (X,τ) is R0 if only if τ is a Wallman basis of (X,τ).
Also (X,τ) is regular if only if τ admits a regular Wallman basis. It is also
clear that every R1 space is R0 and every regular or Hausdorff space is R1.

Theorem 2.11. Let B be an everywhere basic subset of an R1 topological space
(X,τ) such that B 6= X. Then B is compact.

Proof. According to Theorem (2.9), it is enough to prove that Fr(B) = ∅.
Assume, on the contrary, there exists a point p ∈ Fr(B). Define B1 = {V ∈
τ : p 6∈ V} and B2 = {W ∈ τ : p ∈ W}. If B = {V ∪W : V ∈B1 and W ∈B2}
it is clear that B is an annular basis of (X,τ). Observe that for every T =
V ∪W ∈B, we have p 6∈ Fr(T) (because Fr(T) ⊆ Fr(V )∪Fr(W) ⊆ X\{p}).
This implies that B 6∈ B, contradicting the fact that B is everywhere basic. �

Definition 2.12. A topological space (X,τ) is irreducible if every non-empty
open set V ∈ τ is dense in X. Equivalently, (X,τ) is irreducible if every pair
of non-empty open subsets of X have a non-empty intersection.

Theorem 2.13. Let B 6= X be an everywhere basic subset of a topological
space (X,τ). If X \B is irreducible, then B is compact.

Proof. Let U be an open cover of B. Let B′ be the family of open sets L ∈ τ
which are contained in a finite union of members of U and let B′′ = {∅}∪{M ∈
τ : M \B 6= ∅}. Clearly B = {L∪M : L ∈ B′ y M ∈ B′′} is an annular basis
of (X,τ). However, B 6∈ B, a contradiction. �

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A. Garćıa-Máynez and A. Pimienta Acosta

Theorems (2.11) and (2.13) have the following consequences:

Corollary 2.14. An R1 topological spaces (X,τ) is minimally basic iff (X,τ)
is locally compact and 0-dimensional.

Corollary 2.15. Let (X,τ) be an unibasic space and let x ∈ X. Then X\{x}
is compact. Therefore, if X has a compact, closed and non-empty subspace,
then X itself is compact.

Corollary 2.16. Every R1 unibasic space (X,τ) has a finite topology. In fact,
for every x ∈ X, {x} is open and X is a finite union of point-closures.

Definition 2.17. Let (X,τ) be an R0 topological space.
a) (X,τ) is R′1 if every compact open subset of X is closed.
b) (X,τ) is R′′1 every intersection of compact open subspaces of X is

compact.

Remark 2.18. R1 ⇒R′1 ⇒R′′1 ⇒R0.

Proof. (R1 ⇒R′1) It enough to observe that if (X,τ) is R1, K ⊆ X is compact,
V ⊆ X is open and K ⊆ V , then K ⊆ V . �

A subset S of X is a semi-block of a entourage E of X if S ×S ⊆ E.

Lemma 2.19. Let R be a transitive entourage of a set X; let x ∈ X and let
A ⊆ X be a semi-block of R intersecting R(x). Then A ⊆ R(x).

Proof. Select a point y ∈ A ∩ R(x) and let z ∈ A. Therefore, (x,y) ∈ R and
(y,z) ∈ A × A ⊆ R. Since R is transitive, we deduce that (x,z) ∈ R, i.e.,
z ∈ R(x). �

Definition 2.20. Let α be a cover of a set X. For x ∈ X, define Cost(x,α) =⋂
{L: x ∈ L ∈ α}. The indexed cover {Cost(x,α) : x ∈ X} is denoted as α∇

and is called the cobaricentric cover of α. Let α be any cover of a set X. Then
the entourage E(α∇) of the cobaricentric cover α∇ is a transitive entourage of
X.

A cover α of a topological space (X,τ) is interior-preserving if for each
x ∈ X, Cost(x,α) is a τ-neighborhood of x.

Lemma 2.21. Let R be a totally bounded transitive entourage on a set X.
Then the family {L: L = R(x) for some x ∈ X} is finite.

Proof. Let {A1,A2, . . . ,An} be a finite cover of X consisting of semi-blocks of
R. By Lemma (2.19), each R(x) is the union of the sets Ai which intersect R(x).
Hence the family {L: L = R(x) for some x ∈ X} has at most 2n elements. �

Theorem 2.22. Let (X,τ) be a topological space. Consider the following prop-
erties:

(1) τ is finite.
(2) P is the only quasi-uniformity on X which induces τ.
(3) Every interior-preserving cover of X is finite.

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A note on unibasic spaces and transitive quasi-proximities

(4) (X,τ) is hereditarily compact.
(5) δP is the only quasi-proximity on X which induces τ.
(6) δP is the only transitive quasi-proximity on X which induces τ.
(7) (X,τ) is unibasic.

Then 1) ⇒ 2) ⇒ 3) ⇒ 4) ⇒ 5) ⇒ 6) ⇒ 7); if (X,τ) is R′′1 , 7) ⇒ 4) and if
(X,τ) is R′1, 7) ⇒ 1).

Proof. The proofs of the implications 1) ⇒ 2) ⇒ 3) ⇒ 4) ⇒ 5) appear in
([1]). However, using Lemma (2.21) we obtain a quick proof of the implication
2) ⇒ 3). Assuming 2), we deduce that P = FT . Hence, if α is an interior-
preserving cover of X, the entourage R = E(α∇) is totally bounded and transi-
tive. Therefore, by Lemma (2.21), the family {L: L = R(x) for some x ∈ X} is
finite. This, in turn, implies that α is finite. Indeed, consider the topology of X
whose closed sets are arbitrary unions of arbitrary intersections of elements of
α. The point-closures in this topology are precisely the sets Cost(x,α), where
x ∈ X. Since every closed set in this topology is finite we conclude that this
topology is finite and hence, α is finite. The implication 5) ⇒ 6) is evident
and 6) ⇒ 7) is a consequence of Theorem (2.8). If (X,τ) is R′′1 and V ∈ τ,
V 6= X, clearly V is the intersection of all the compact open sets X\{x}, where
x ∈ X \ V . By hypothesis, V must be compact. We have proved then that
7) ⇒ 4) when (X,τ) is R′′1 -space. Finally, if (X,τ) is R′1, each set X \{x} is
compact and open and, hence, it is also closed. Therefore, each point-closure
is open. Since X is compact, X is the closure of a finite subset of X. Since
(X,τ) is R0, the topology τ must be finite. �

H.-P. Künzi has proved that properties 3), 4), 5), 6), 7) and

2’) P is the only totally bounded quasi-uniformity on X which induces τ
are equivalent (see [4]).

The validity of the implication 7) ⇒ 2) is still open.
Typical examples of topological spaces admitting a unique totally bounded

quasi-uniformity are the hereditarily compact spaces and set ω0 equipped with
the lower topology {[0,n] : n ∈ ω0}∪{∅,ω0}.

The space with carrier set ω0 + 2 and topology {[0,n] : n ∈ ω0}∪{(ω0 + 2)\
{ω0 + 1},ω0 + 2, (ω0 + 2) \ {ω0},∅} admits a unique totally bounded quasi-
uniformity, while this is not true for its subspace (ω0 + 2) \{ω0} (see example
page 148 [4]).

Example 2.23 (see example 1 in [5]). Let N be the set of the positive integers
equipped with the topology τ = {{1, . . . ,n}: n ∈ N}∪ {∅,N}. Obviously,
every proper open subset of N is compact, but N is not compact. This example
shows that a topological space that admits a unique compatible quasi-proximity
need not be compact.

Question: If (X,τ) is an unibasic space is equivalently to say the P is the only
compatible quasi-uniformity?

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A. Garćıa-Máynez and A. Pimienta Acosta

References

[1] P. Fletcher and W. Lindgren, Quasi-uniformity spaces, vol 77, Marcel Dekker, Inc., New
York, First edition, 1982.

[2] J. Ferrer, On topological spaces with a unique quasi-proximity, Quaest. Math 17 (1994),

479–486.
[3] H.-P. Künzi, Nontransitive quasi-uniformities in the Pervin quasi-proximity class, Proc.

Amer. Math. Soc. 130 (2002), 3725–3730.

[4] H.-P. Künzi, Quasi-uniform spaces-eleven years later, Topology Proceedings 18 (1993),
143–171.

[5] H.-P. Künzi, Topological spaces with a unique compatible quasi-proximity, Arch. Math.
43 (1984), 559–561.

[6] H.-P. Künzi and M. J. Pérez-Peñalver, The number of compatible totally bounded quasi-

uniformities, Act. Math. Hung. 88 (2000), 15–23.
[7] H.-P. Künzi and S. Watson, A nontrivial T1-spaces admitting a unique quasi-proximity,

Glasg. Math. J. 38 (1996), 207–213.

[8] W. J. Pervin. Quasi-uniformization of topological spaces, Math. Annalen 147 (1962),
116–117.

[9] W. J. Pervin, Quasi-proximities for topological spaces, Math. Annalen 150 (1963), 325–

326.

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