Download ATOMS IN THE LATTICE OF QUASI-UNIFORMITIES 1. Introduction

The Extended Abstracts of
The 4th Seminar on Functional Analysis and its Applications
2-3rd March 2016, Ferdowsi University of Mashhad, Iran
ATOMS IN THE LATTICE OF QUASI-UNIFORMITIES
BEHNAM BAZIGARAN, MEHRNAZ POURATTAR1∗
1
Department of Mathematics,University of Kashan, Kashan, Iran.
[email protected]
2
Department of Mathematics,University of Kashan, Kashan, Iran.
[email protected]
Abstract. In this paper, we review the properties of atoms in
the lattice of quasi-uniformities. In particular, we shall show that
in the lattice of quasi-uniformities a quasi-uniformity is an atom
if and only if it generated by special preorders. Furthermore, we
observe that any atom in the lattice of quasi-uniformity must be
totally bounded and transitive while it cannot be a uniformity.
1. Introduction
The theory of uniformity space was first introduced by Weil in 1937
to expand metric spaces. Indeed the uniform spaces are topological
spaces that equipped with a structure in order to study some uniform
properties like uniform countinous, uniform convergent, complementry
in them. After Weil many studies have been done on these structures
and by removing some of the conditions in the definition of a uniform
spaces, weakly structures were raised that have many properties of uniform spaces which we can note the study of Nachbin on quasi-uniform
spaces and the research of Fletcher on local quasi-uniform spaces. In
this article, first we introduce some basic definition of quasi-uniform
2010 Mathematics Subject Classification. Primary 47A55; Secondary 39B52,
34K20, 39B82.
Key words and phrases. Quasi-uniformity, Atom, Complete lattice.
∗
Speaker.
1
2
BAZIGARAN, POURATTAR
spaces.
Let X be a nonempty set. A filter U on X × X is called a quasiuniformity on X provided that:
(1) Each member of U is a reflexive relation;
(2) For each V ∈ U there exists U ∈ U such that U ◦ U ⊆ V .
Example 1.1. If we consider the singleton filter I = {X × X}, then I
is a uniformity that is called indiscrete uniformity. Also if we assume
that D is a filter that consists all of reflexive relation on X, then D is
a uniformity that is called discrete uniformity.
Proposition 1.2. If U be a quasi-uniformity on a set X, then the
inverse of U that defines by
U −1 = {U −1 : U ∈ U},
is also a quasi-uniformity on X.
We note that a quasi-uniformity U is called a uniformity if U = U −1 .
Remark 1.3. If B ⊆ P(X) be a filter base on X, then we demonstrate
the filter generated by B via f il{B} that define by
f il{B} = {F ⊆ X | ∃ E ∈ B : E ⊆ F }.
Definition 1.4. Consider the quasi-uniformity U on X. The subcollection B from U is called a base for U if every member of U contains
a member of B.
Example 1.5. Let ρ be a quasi-psedu-metric on X. For every ε > 0
we set
Uε = {(x, y) ∈ X × X : ρ(x, y) < ε}.
Then f il{Uε : ε > 0} is a quasi-uniformity on X.
Theorem 1.6. Let X be a set and U be a quasi-uniformity on it. Then
the collection
T (U) = {A ⊆ X : ∀a ∈ A, ∃ U ∈ U : U [a] ⊆ A},
is a topology on X [3].
We consider the set of all quasi-uniformities on set X, partially ordered under set-theoretic inclusion ⊆. It is easy to see that (Q(X), ⊆)
is a complete lattice. In particular we have the following properties:
Proposition 1.7. (Q(X), ⊆) is a complete lattice.
ATOMS IN THE LATTICE OF QUASI-UNIFORMITIES
3
Proof. For every collection {Ui }i∈I , we define:
∪
B={
Ui : Ui ∈ Ui , I0 is finite}.
i∈I0
Then we can see that
∨
Ui = f il{B} and therefore the collection
i∈I
f il{B} is an Upper bound for {Ui }i∈I which deduce that (Q(X), ⊆)
is a complete lattice.
□
Corollary 1.8. Let (U (X), ⊆) be the set of all uniformities on the set
X that partially ordered under set-theoretic inclusion. Then (U (X), ⊆)
is a sub-lattice of (Q(X), ⊆).
Definition 1.9. A quasi-uniformity U on a set X is called totally
bounded provided that for each U ∈ U there exist a finite cover A of
X such that A × A ⊆ U whenever A ∈ A.
In particular, for a given quasi-uniformity U the finest totally bounded
quasi-uniformity coarser than U will be denoted by Uω .
2. Main Results
We recall that a quasi-uniformity A ̸= I on a set X is an atom of
the lattice (Q(X), ⊆) if it is an upper neighbour of I. In other words,
for any quasi-uniformity U on X such that I ⊆ U ⊆ A, we have I = U
or U = A.
It is clear that if A is an atom, then A−1 is an atom too. It is significant
that if A is an atom of (Q(X), ⊆), then A = Aω . In the other hand
we have the following propsition:
Proposition 2.1. Each atom of (Q(X), ⊆) is totally bounded.
The characterization below shows that any atoms in the (Q(X), ⊆),
have a base that is a singleton and so are transitive.
Remark 2.2. In the following of our paper we always consider
T (A, B) = (X × X) \ (A × B).
It is easy to see that
T (A, B) = [(X \ A) × X] ∪ [X × (X \ B)].
Lemma 2.3. Let A be a proper subset of X. Then A = f il{T (A, X \
A)} is a transitive atom of the lattice (Q(X), ⊆) [1].
Theorem 2.4. A quasi-uniformity A of X is a transitive atom of
(Q(X), ⊆) if and only if there exists a nonempty subset A of X such
that A = f il{T (A, X \ A)} [1].
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BAZIGARAN, POURATTAR
The other noticable remark is that if A is an atom of (Q(X), ⊆), then
A can not be a uniformity. To prove this, first we have the following
proposition:
Proposition 2.5. Let X be a set and A ⊆ X. Then
(1) (T (A, X \ A))−1 = T (X \ A, A);
(2) (f il{T (A, X \ A)})−1 = f il{T (X \ A, A)}.
Corollary 2.6. If A is an atom of (Q(X), ⊆), then it cannot be a
uniformity. Furthermore no uniformity of (Q(X), ⊆) can be an atom.
Example 2.7. There does not exist any atom of (Q(R), ⊆) that is
coarser than the usual uniformity R on the set R of the reals. Indeed,
if in this lattice, assume A such that coarser than R, then obviuosly A
is a uniformity that according to (2.6), cannot be an atom.
The following example express that the converse of proposition (2.1)
is generally not true. In other words every totaly bounded quasiuniformity on an arbitrary set, it is not nesessary that be an atom
in the lattice of quasi-uniformity on that set.
1
: n ∈ N} and for each x, y ∈ X
n+1
define d(x, y) = |x − y|. Then (X, d) is a metric space that results a
uniformity such Ud . Since (X, d) is bounded and X ⊆ R, it is easy to
see that it is totally bounded too. So we can see that Ud is also totally
bounded. But Because Ud is a uniformity, it cannot be an atom.
Example 2.8. We set X = {
Proposition 2.9. All atoms of (Q(X), ⊆) are transitive [1].
Corollary 2.10. if X is a singletone, no atoms of (Q(X), ⊆) exist. If
X is a finite set with at least 2 elements then (Q(X), ⊆) has 2|X| − 2
atoms. If X is an infinite set, then (Q(X), ⊆) has 2|X| atoms.
References
1. E. P. De Jager, H. P. A. Kunzi, Atoms, anti-atoms and other complements in
the lattice of quasi-uniformities, Topology Appl., 153(2006), no. 16, 3140-3156.
2. E. P. De Jager, H. P. A. Kunzi, The lattice of quasi-uniformities, Elsevier Science,
2007.
3. P. Fletcher and W. F. Lindgren, Quasi-uniform spaces, Marcel Dekker Inc., 1982.
4. H. P. A. Kunzi, An introduction to quasi-uniform spaces, Contem. Math.,
486(2009), 239-304.