Strongly Pre-irresoluteness and Weakly Pre-irresoluteness Based on Continuous Valued Logic

Abstract

In this paper, we present and study the concepts of fuzzifying pre-$\theta$-neighborhood system of a point, fuzzifying pre-$\theta$-closure of a set, fuzzifying pre-$\theta$-interior of a set, fuzzifying pre-$\theta$-open sets and fuzzifying pre-$\theta$-closed sets in fuzzifying topological spaces. The basic properties of these concepts are investigated. Two types of functions in a fuzzifying topological spaces called fuzzifying strongly pre-irresolute and fuzzifying weakly pre-irresolute functions are introduced. Then the interrelations of these functions with the parallel existing allied concepts are established. Finally, several characterizations of these functions along with different conditions for their existence are obtained.

Appendix A

Example 3.7

${\mathscr {P}}(X)$	${\mathscr {T}}$	${\mathscr {T}}_{p}$
$\emptyset$	1	1
X	1	1
$\{x\}$	1	1
$\{y\}$	0	0

Example 3.21

${\mathscr {P}}(X)$	${\mathscr {T}}$	${\mathscr {T}}_{p}$	${\mathscr {T}}_{p\theta }$
$\emptyset$	1	1	1
X	1	1	1
$\{x\}$	1	1	$\frac{1}{6}$
$\{y\}$	0	0	1
$\{z\}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{6}$
$\{x,y\}$	0	$\frac{5}{6}$	1
$\{x,z\}$	1	1	$\frac{1}{6}$
$\{y,z\}$	$\frac{1}{6}$	$\frac{1}{6}$	1

Example 4.5

${\mathscr {U}}(X)$	${\mathscr {U}}$	${\mathscr {U}}_{p}$
$\emptyset$	1	1
X	1	1
$\{x\}$	0	1
$\{y\}$	0	1
$\{z\}$	0	0
$\{x,y\}$	1	1
$\{x,z\}$	0	1
$\{y,z\}$	0	1

Example 4.7

${\mathscr {U}}(X)$	${\mathscr {U}}$	${\mathscr {U}}_{p}$
$\emptyset$	1	1
X	1	1
$\{x\}$	0	1
$\{y\}$	1	1
$\{z\}$	0	1
$\{x,y\}$	0	1
$\{x,z\}$	0	0
$\{y,z\}$	0	1