Approach Merotopological Spaces and their Completion
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 409804, 16 pages
doi:10.1155/2010/409804
Research Article
Approach Merotopological Spaces and
their Completion
Mona Khare and Surabhi Tiwari
Department of Mathematics, University of Allahabad, Allahabad 211002, India
Correspondence should be addressed to Mona Khare,
Received 24 July 2009; Accepted 13 April 2010
Academic Editor: Richard Wilson
Copyright q 2010 M. Khare and S. Tiwari. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
This paper introduces the concept of an approach merotopological space and studies its categorytheoretic properties. Various topological categories are shown to be embedded into the category
whose objects are approach merotopological spaces. The order structure of the family of all
approach merotopologies on a nonempty set is discussed. Employing the theory of bunches, bunch
completion of an approach merotopological space is constructed. The present study is a unified
look at the completion of metric spaces, approach spaces, nearness spaces, merotopological spaces,
and approach merotopological spaces.
1. Introduction
Some of the applications of nearness-like structures within topology are unification,
extensions, homology, and connectedness. The categories of R0 -topological spaces, uniform
spaces 1, 2, proximity spaces 2, 3, and contiguity spaces 4, 5 are embedded into the
category of nearness spaces. The study of proximity, contiguity, and merotopic spaces in
the more generalized setting of L-fuzzy theory can be seen in 6–13. In 14, the notion of
an approach space was introduced via different equivalent set of axioms to measure the
degree of nearness between a set and a point. While developing the theory of approach
spaces, Lowen et al. many a time employed tools from nearness-like structures. The notion of
“distance” in approach spaces is closely related to the notion of nearness; further proximity
and nearness concepts arise naturally in the context of approach spaces as can be seen in 15–
18. Hence it became mandatory looking into the nearness-like concepts in approach theory,
more clearly. With the same spirit, Lowen and Lee 19 made an attempt to measure how near
a collection of sets is and, in the process, axiomatized the two equivalent concepts: approach
merotopic and approach seminearness structures, respectively, to measure the degree of
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smallness and nearness of an arbitrary collection of sets, and therefore generalized approach
spaces in a sense. In 2004, Bentley and Herrlich 20 gave the idea of merotopological
spaces as a supercategory of many of the above mentioned categories. They also constructed
the functorial completion of merotopological spaces employing the theory of bunches in
merotopological spaces. In 21, we axiomatized the notion of approach nearness by adding
to the axioms of an approach merotopy the axiom relating a collection of sets and the
closure induced by the respective approach merotopy; and we analogously obtained cluster
completion of an approach nearness space.
Prerequisites for the paper are collected in Section 2. In Section 3, we axiomatize
approach merotopological spaces. The category AMT of approach merotopological spaces
and their respective morphisms is shown to be a topological construct and a supercategory
of some of the known topological categories, including the category of topological spaces
and continuous maps. Order structure of the family of all approach merotopologies on X
is also discussed. In Section 4, bunch completion of an approach merotopological space is
constructed, employing the theory of bunches. The concept of regularity in an approach
merotopic space is introduced to obtain a relationship between cluster and bunch completion
of an approach nearness space. Indeed, it is shown that cluster completion is a retract of the
bunch completion of a regular approach nearness space X, ν.
2. Preliminaries and Basic Results
Let X be a nonempty ordinary set. The power set of X is denoted by PX and the family of
all subsets of PX is denoted by P2 X. We denote by ℵ0 the first infinite cardinal number,
by |A| the cardinality of A where A ⊆ X, and by J an arbitrary index set. For A, B subsets of
PX, A ∨ B ≡ {A ∪ B : A ∈ A, B ∈ B}; A corefines B written as A ≺ B if and only if for all
A ∈ A there exists B ∈ B such that B ⊆ A. For A ⊆ PX, stackA {A ⊆ X : B ⊆ A for some
B ∈ A} and sec A {B ⊆ X : A ∩ B /
∅, for all A ∈ A} {B ⊆ X : X − B /
∈ stackA}. Observe
∈ G;
that sec2 A stack A, for all A ∈ P2 X. A grill on X is a subset G of PX satisfying ∅ /
if A ∈ G and A ⊆ B, then B ∈ G; and if A ∪ B ∈ G, then A ∈ G or B ∈ G. For basic definitions
and results of merotopic spaces and nearness spaces, we refer to 1.
Definition 2.1 see 20. A merotopological space is the triple X, ξ, cl, where ξ is a merotopy
and cl is a Kuratowski closure operator on X such that {clA : A ∈ A} ∈ ξ ⇒ A ∈ ξ, for all
A ∈ P 2 X.
Definition 2.2 see 19, 21. A function ν : P2 X → 0, ∞ is called an approach merotopy on
X if for any A, B ∈ P2 X the following conditions are satisfied:
AM1 A ≺ B ⇒ νA ≤ νB,
AM2 A
/ ∅ ⇒ νA 0,
AM3 ∅ ∈ A ⇒ νA ∞,
AM4 νA ∨ B ≥ νA ∧ νB.
The pair X, ν is called an approach merotopic space. For an approach merotopic space X, ν,
we define clν A {x ∈ X : ν{{x}, A} 0}, for all A ⊆ X. Then clν is a Čech closure
operator on X.
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An approach merotopy ν on X is called an approach nearness on X 21 if the following
condition is satisfied:
AN5 ν{clν A : A ∈ A} ≥ νA.
In this case, clν is a Kuratowski closure operator on X. Denote clA {clA : A ∈ A}.
Definition 2.3 see 21. Let A ∈ P2 X and ν be an approach merotopy on X. Then we say
that A is
i near in ν if νA 0,
ii ν-clan if A is a near grill,
iii ν-closed if ν{{A} ∪ B} 0, for all B ⊆ A ⇒ A ∈ A,
iv ν-cluster if A is a ν-closed ν-clan.
For any approach merotopic spaces X, νX and Y, νY , a map f : X → Y is called a
contraction if νY fA ≤ νX A, for all A ∈ P2 X, or equivalently νX f −1 B ≥ νY B, for
all B ∈ P2 Y . For any approach merotopies ν1 and ν2 on X, ν2 ≤ ν1 ν1 is finer than ν2 , or ν2 is
coarser than ν1 if the identity mapping 1X : X, ν1 → X, ν2 is a contraction see 19. For
standard definitions in the theory of categories we refer to 22, for approach spaces we refer
to 14, and for lattices see 23.
3. Approach Merotopological Spaces
In this section, we introduce approach merotopological spaces and establish some categorytheoretic results for them. Lattice structure o (...truncated)
