COVERS IN THE LATTICE OF FUZZY TOPOLOGIES – I

Sunil C. Mathew and T. P. Johnson

Abstract. In this paper we combine the concept of simple extensions of a fuzzy topology with closure structure of fuzzy points to give certain necessary and sufficient conditions for a fuzzy topology to have a cover in the lattice of fuzzy topologies.

1. Introduction. The concept of covers in the lattice of topologies and consequently that of adjacent topologies was introduced by Pushpa Agashe and Norman Levine in [1]. In this paper we study the concept of covers in the lattice of fuzzy topologies and consequently that of adjacent fuzzy topologies. In [3] the authors investigated certain classes of fuzzy points based on the structure of their closure. In this note we establish how these classes of fuzzy points together with the idea of simple extensions give rise to certain necessary and sufficient conditions for a fuzzy topology to have a cover.

2. Preliminaries. In this section, we include certain definitions needed for the subsequent development. We denote by I_{d}, the discrete fuzzy topology in the lattice τ_{d} of all fuzzy topologies on a set X.

\underline{Definition 2.1}. Let (X, F) be an fts. A fuzzy set g in X is said to be generalized closed (g–closed) if and only if cl g ≤ f whenever g ≤ f and f ∈ F (cl g denotes the smallest closed fuzzy set containing g).

\underline{Definition 2.2}. Let (X, F) be an fts. A fuzzy set g in X is said to be nearly crisp if cl g ∧ ℂ(cl g) = \underline{0}.

\underline{Definition 2.3}. Let (X, F) be an fts. A fuzzy point x_{λ} in X is said to be well–closed if cl x_{λ} is again a fuzzy point in X.

\underline{Definition 2.4}. Let (X, F) be an fts. Then it is called a c–fts if the members of F are characteristic functions.

3. Simple Extensions and Covers.

\underline{Definitions 3.1} [3]. Let (X, F) be an fts and suppose that g ∈ I^{X} and g ∉ F. Then the collection F (g) = {g_{1} ∨(g_{2} ∧g) ; g_{1}, g_{2} ∈ F } is called the simple extension of F determined by g.

\underline{Theorem 3.2} [3]. Let (X, F) be an fts and suppose that F (g) is the simple extension of F determined by g. Then

(i) F (g) is an fts on X and (ii) F (g) = F ∨ G where G = \{\underline{0}, \underline{1}, g\}.

\underline{Definition 3.3}. Let F and G be two fuzzy topologies on a set X. Then F imp G if and only if (i) F < G and (ii) if F ≤ H ≤ G where H is an fts on X, then either F = H or H = G. We call F and G adjacent if and only if F imp G or G imp F. Also if F imp G then we say that F is an immediate predecessor of G and G is an immediate successor of F or G is a cover of F.

\underline{Theorem 3.4} [3]. Let F and G be two fuzzy topologies on a set X such that F imp G. Then G is a simple extension of F.

\underline{Theorem 3.5}. I_{d} is not a cover of any fuzzy topology on a set X.

\underline{Proof}. To prove this theorem it suffices to show that for any g ∉ F, F (g) ≠ I_{d}. Given any g ∉ F, we can find fuzzy subsets f_{1}, f_{k}, g_{1}, g_{k} ≠ g such that f_{1} ∨ f_{k} = g = g_{1} ∧ g_{k} with f_{1}, g_{1} ∉ F. Now repeating this argument on g with f_{1}, g_{1} and subsequent fuzzy subsets we get a sequence of fuzzy subsets as follows:

⋯ ≤ f_{2} ≤ f_{1} ≤ g ≤ g_{1} ≤ g_{2} ≤ ⋯

where f_{i}, g_{i} ≠ g and f_{i}, g_{i} ∉ F, for every i. Also by the denseness property of real numbers, it follows that the choice of f_{i} and g_{i} for any i is uncountable. Hence, we must have F (g) ≠ I_{d}.

As a consequence of the above theorem, we get the following result in [2].

\underline{Corollary 3.6}. There is no dual atom in τ_{d}.

In [3], the following theorem was proved.

\underline{Theorem 3.7}. Let (X, F) be a c–fts and let x_{γ}; γ = 1 be a closed fuzzy point in X. Then for any fuzzy point x_{λ}, F (ℂ(x_{λ})) is a cover of F.

In the above theorem, F was restricted to a c–fts. But in the following theorem we remove that restriction and prove the result for any arbitrary fts.

\underline{Theorem 3.8}. Let (X, F) be any fts in which the fuzzy point x_{γ}; γ = 1 is closed. Then for any fuzzy point x_{λ}, F (ℂ(x_{λ})) is a cover of F.

\underline{Proof}. Let U be any fts on X such that F ≤ U ≤ F (ℂ(x_{λ})); F ≠ U. Consider any f ∈ F (ℂ(x_{λ})) such that f ∈ U − F. Then f = f_{1} ∨ (f_{2} ∧ ℂ(x_{λ})); f_{1}, f_{2} ∈ F.

Since x_{γ}; γ = 1 is closed, we have f ∨ ℂ(x_{γ}) = ℂ(x_{λ}) ∈ U. Then by Theorem 3.2 we get U = F (ℂ(x_{λ})) and hence, F (ℂ(x_{λ})) is a cover of F.

The following theorem gives a necessary condition for a fuzzy topology to have a cover.

\underline{Theorem 3.9}. Let (X, F) be an fts such that every g ∉ F contains a g–closed and nearly crisp fuzzy point x_{λ}; λ ≠ 1 with g(x) = λ. If for each such g, there exists an h ∈ F containing x_{λ} with h ∨ g ∉ F, then F has no covers.

\underline{Proof}. To prove that F has no covers, we will show that for any g ∉ F, the simple extension F (g) is not a cover of F. Since g ∉ F, there exists a g–closed and nearly crisp fuzzy point x_{λ}; λ ≠ 1 with g(x) = λ. Also, we can find an h ∈ F containing x_{λ} such that g ∨ h ∉ F. Then we must have F < F (g ∨ h) ≤ F (g).

If possible, let g ∈ F (g ∨ h). Then g = f_{1} ∨ (f_{2} ∧ (g ∨ h)); f_{1}, f_{2} ∈ F. Then either x_{λ} ∈ f_{1} or x_{λ} ∉ f_{1}. Since x_{λ} is g–closed and nearly crisp, in both cases we get g(x) = 1, a contradiction. Thus, we get F < F (g ∨ h) < F (g). Then by Theorem 3.4, F has no covers.

The following theorem gives a sufficient condition for a fuzzy topology to have a cover.

\underline{Theorem 3.10}. Let (X, F) be an fts. If it contains a well–closed but not closed fuzzy point, then F has a cover.

\underline{Proof}. Let x_{λ} be a well–closed but not closed fuzzy point in (X, F). We claim that F (ℂ(x_{λ})) is a cover of F. Suppose that F ≤ G ≤ F (ℂ(x_{λ})); F ≠ G. Let g ∈ G−F. Then g = g_{1} ∨(g_{2} ∧ℂ(x_{λ})); g_{1}, g_{2} ∈ F. Define h = ℂ(cl x_{λ}) ∈ G so that h ∨ g = ℂ(x_{λ}) ∈ G. Then by Theorem 3.2, G = F (ℂ(x_{λ})) and hence, F (ℂ(x_{λ})) is a cover of F.

\underline{References}

1. P. Agashe and N. Levine, “Adjacent Topologies,” J. Math. Tokushima Univ., 7 (1973), 21–35.

2. T. P. Johnson, “On the Lattice of Fuzzy Topologies I,” Fuzzy Sets and Systems, 48 (1992), 133–135.

3. T. P. Johnson and Sunil C. Mathew, “On Fuzzy Points in Fuzzy Topological Spaces,” FJMS, sp. vol. (2000), 75–86.

4. N. Levine, “Simple Extensions of Topologies,” American Mathematical Monthly, 71 (1964), 22–25.

Sunil C. Mathew Department of Mathematics St. Thomas College Pala Arunapuram P.O., 686 574 Kottayam (Dt), Kerala INDIA email: sunilcmathew@rediffmail.com

T. P. Johnson Cochin University College of Engineering Kuttanad, Pulincunnu P. O. Alleppey 688 504 Kerala, INDIA email: johnsontp@rediffmail.com