We define the concepts of -metric in sets over -complete Boolean algebra and obtain some applications of them on the theory of topology. We also study some related properties of them.
1. Introduction
Numerous studies have been made concerning geometries and topologies induced in sets by general distance functions. A formulation of the notion βgeneralized metric space (or -metric space)" has been given [1]. In this paper, we begin the elaboration of the topology induced in sets over -complete Boolean algebra.
In this paper, shall always denote a -complete Boolean algebra. In , we denote the operations of join, meet, and complement by , and , respectively.
2. The -Metric Spaces
In 1952, a new structure of metric spaces, so called -metric space was introduced by Ellis and Sprinkle [2], on the set to Boolean algebra (for details see [3, 4]).
Definition 2.1 (see [2]). A -metric space is a set with a map ( is -Boolean algebra) with the properties(1) if and only if ,(2), (symmetry), and (3), for all , , belong to .
In [1], the present author has introduced a new structure of metric spaces which is a generalized idea of the ordinary metric space. The term generalized metric space is used in [5, 6].
Definition 2.2 (see [7]). Let be a nonempty set and be a function satisfying the following properties:() if ,(); for all , with ,(), for all with ,(), (symmetry in all three variables), and, for all , (rectangle inequality).
2.1. Topology Of -Metric Spaces
In this section we define generalized -metric (or -metric) space and introduce some basic notions and results that are used in sequel.
Definition 2.3. Let be a non empty set and , be a function satisfying the following properties() if ,();ββ forββall , with ,(),ββforββall with ,(), (symmetry in all three variables), andforall , (rectangle inequality).Then, the function is called a generalized -metric, or, more specifically, a -metric on , and the pair is called a -metric space.
Example 2.4. Put . Define a map by
The map is a -metric onββ.
Remark 2.5. We can show that a -metric space is a generalized -metric space over .
Proposition 2.6. Every -metric space will define a -metric by .
Proof. Conditions 1 and 2 of -metric are clearly and 3 follows from .
Proposition 2.7. Let be -metric on a ring . For an element , the following maps are -metrics on the ring : (1) and(2).
Proposition 2.8. Let be a -metric space. Define a function, by for . The map is a -metric on , and consequently is a -metric space.
Proof. Conditions , and are clear. We show that and are valid too.
And for , we have
Proposition 2.9. Let be a -metric space, then for all , it follows that: (1),(2),(3), and(4).
Proposition 2.10. Let be a -metric space, then the following are equivalent: (1) for all ,(2) for all .
Definition 2.11. Let be a -metric. For all , , the -ball with center and radius is . For any -ball, we can define .
Proposition 2.12. Let be a -metric space. Then for all and , we have (1)If , then and(2)if , then there exists one element such that, .
Proof. (1) follows directly . Put and use (), then deduce (2).
It follows from (2) of the above proposition that the family of all -balls
is the base of topology on , we call the -metric topology.
Definition 2.13. Let be a -metric space. The sequence is -convergent to if it converges to in the -metric topology, .
Remark 2.14. Another topological notions as -Hausdorff, -compact, -normal, and β¦ define similarly as usual.
Definition 2.15. Let and be -metric spaces, a function is -continuous at a point if for all . We say is -continuous if it is -continuous at all points of , that is, continuous as a function from with the -topology to with the -topology.
Theorem 2.16. Let be a -metric space and be a self map of into itself. Suppose is -continuous at , if there is a point such that the sequence of iterates converges to , then .
Proof. From , we derive as we deduce, ,, so the above equation will be , therefore, .
Theorem 2.17. Let be a -metric space and be -continuous. If for , the sequence has a convergent subsequence -converges to , and , then has a fixpoint.
Proof. Since , then by -continuity of we have , and we have
As we get , which implies that .
Theorem 2.18. A -metric space is -Hausdorff.
Proof. Consider two elements , belong to that , and two subsets and . These are two disjoint neighbourhoods of and , respectively. It is sufficient to prove that the subsets and are open. Suppose . Put , and . If , then , therefore, and . This proves is Hausdorff.
Theorem 2.19. A -metric space is -normal.
Proof. Let . Consider two closed disjoint subsets and of . Put and . It is clear that are disjoint open subsets of , , and .
Corollary 2.20. A -metric space is -regular.
2.2. Monoid Invariant -Metric
Definition 2.21. A -metric on a is -invariant (or monoid invariant) when and are valid for any .
Proposition 2.22. Let be a monoid. If is an m-invariant -metric on X, then (1) and(2).
Proof. (1) We have (2) It concludes by induction.
Put is a monoid and be a -metric on . We call is left effective on if for all then . And we say is left free -metric on if for a triple , then . The right effective (free) is defined samely, and we call is effective (free) if it is left and right effective (free) on . Immediately we deduce, if is free -metric on then it is effective -metric.
Proposition 2.23. Put is a monoid homomorphism and is a -metric on the following results are valid: (1), is a -metric on .(2)If is m-invariant, then is m-invariant.(3)The -metric on is free (or effective) if is free (or effective) on .
Let and be subsets of . The set is congruent to , written , provided there exist a map that for any , and that is congruent to by the map . The map is called a congruence between and . Clearly every single set , and each map , congruent to .
Definition 2.24. A motion of is a congruence of with itself.
Proposition 2.25. Let be a -metric space. The set is a group.
Proof. The identity map belongs to so is not empty. Group action is composition of maps, so the identity map is the identity element. If , then , so .
Theorem 2.26. Let be a -metric space and is a motion of , with the properties: if , then , then the set of fixpoints of (or ) is equal to the set of fixpoints of for all .
Proof. Clearly . Suppose , so the sequence , because , for all . Now by Theorem 2.16, we conclude .
Proposition 2.27. Let be a monoid and be a -metric on . (1)If for all , and be motions of , then is m-invariant.(2) (or ) () cannot be a motion of , if is effective.
Example 2.28. The -metric defined in Example 2.4 is invariant on the monoid , but it is not effective. Clearly for all , , is a motion.
References
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