SUPRA D−SETS AND ASSOCIATED SEPARATION AXIOMS Jamal
... Definition 2.11. A supra topological space (X, µ) is called a supra symmetric space if for x and y in X, x ∈ Clµ ({y}) implies y ∈ Clµ ({x}). Theorem 2.12. Let (X, µ) be a supra symmetric space. Then the following are equivalent: (1) (X, µ) is S − T0 ; (2) (X, µ) is S − T1 . Proof. It is enough to s ...
... Definition 2.11. A supra topological space (X, µ) is called a supra symmetric space if for x and y in X, x ∈ Clµ ({y}) implies y ∈ Clµ ({x}). Theorem 2.12. Let (X, µ) be a supra symmetric space. Then the following are equivalent: (1) (X, µ) is S − T0 ; (2) (X, µ) is S − T1 . Proof. It is enough to s ...
Some kinds of fuzzy connected and fuzzy continuous functions
... If 1X A B where A B 0 X , A and B are non-empty fuzzy open sets of X , then they are complements to each other ,and hence both are fuzzy open and fuzzy closed set . 1.14 Definition[A.M.Zahran,2000] A family of fuzzy sets is called a cover of a fuzzy set A if and only if A {Bi : Bi } ...
... If 1X A B where A B 0 X , A and B are non-empty fuzzy open sets of X , then they are complements to each other ,and hence both are fuzzy open and fuzzy closed set . 1.14 Definition[A.M.Zahran,2000] A family of fuzzy sets is called a cover of a fuzzy set A if and only if A {Bi : Bi } ...
An Introduction to Simplicial Sets
... simplicial set defined by (SY )n = homTop (|∆n | , Y ), where di and sj are defined by di σ = σdi : |∆n−1 | → Y and sj σ = σsj : |∆n+1 | → Y , for σ ∈ S(Y )n . Note that the relations (8), (9), and (10) for SY follow from di and sj satisfying the opposite relations in ∆. It turns out that |−| and S ...
... simplicial set defined by (SY )n = homTop (|∆n | , Y ), where di and sj are defined by di σ = σdi : |∆n−1 | → Y and sj σ = σsj : |∆n+1 | → Y , for σ ∈ S(Y )n . Note that the relations (8), (9), and (10) for SY follow from di and sj satisfying the opposite relations in ∆. It turns out that |−| and S ...
1. Introduction - Departamento de Matemática
... It is clear that Theorem 1.1 provides an alternative definition of second countability that, in the absence of the axiom of choice, turns out to be non-equivalent to the familiar definition. Starting from these two definitions of second countability, we will discuss the consequences of replacing one ...
... It is clear that Theorem 1.1 provides an alternative definition of second countability that, in the absence of the axiom of choice, turns out to be non-equivalent to the familiar definition. Starting from these two definitions of second countability, we will discuss the consequences of replacing one ...
I-fuzzy Alexandrov topologies and specialization orders
... classical preorder on X, precisely, all upper sets with respect to forms an Alexandrov topology. Conversely, we can also obtain a preorder on X determined by a given topology on X, called a specialization order usually. This process determined one by another had been investigated in the fuzzy se ...
... classical preorder on X, precisely, all upper sets with respect to forms an Alexandrov topology. Conversely, we can also obtain a preorder on X determined by a given topology on X, called a specialization order usually. This process determined one by another had been investigated in the fuzzy se ...
Minimal Totally Disconnected Spaces
... of p in Y}. Then owis a free openfilter on the spaceX. Considerany nondegenerate subsetC of X. SinceC U {p} is not a connectedsubsetof the totally disconnectedspaceY, there exist open subsetsL and M of Y suchthat C U {p} is separatedby L and M, where, sayp CM. If M ClC • 0, then C is separatedin the ...
... of p in Y}. Then owis a free openfilter on the spaceX. Considerany nondegenerate subsetC of X. SinceC U {p} is not a connectedsubsetof the totally disconnectedspaceY, there exist open subsetsL and M of Y suchthat C U {p} is separatedby L and M, where, sayp CM. If M ClC • 0, then C is separatedin the ...
SOME ASPECTS OF TOPOLOGICAL TRANSITIVITY——A SURVEY
... (see Example 3.1.3). Hence, whenever J1 and J2 are nonempty open subintervals of I , there is a periodic orbit of g which intersects both J1 and J2 . This gives (TT) for (X, f ). 2.2. On the equivalent formulations of the definition. Standard dynamical systems. Nevertheless, under some additional as ...
... (see Example 3.1.3). Hence, whenever J1 and J2 are nonempty open subintervals of I , there is a periodic orbit of g which intersects both J1 and J2 . This gives (TT) for (X, f ). 2.2. On the equivalent formulations of the definition. Standard dynamical systems. Nevertheless, under some additional as ...
88 CHAPTER 5 KURATOWSKI CLOSURE OPERATORS IN GTS
... By definition, φ is net closed and hence K(φ) = φ. Ncl(A) is the smallest net closed set containing A and hence A⊂K(A). Since Ncl(A) is net closed, Ncl(Ncl(A)) = Ncl(A) and hence K(K(A) = K(A). A⊂NclA, B⊂NclB hence A∪B⊂NclA∪NclB. Now NclA and NclB are net closed sets and hence NclA∪NclB is a net clo ...
... By definition, φ is net closed and hence K(φ) = φ. Ncl(A) is the smallest net closed set containing A and hence A⊂K(A). Since Ncl(A) is net closed, Ncl(Ncl(A)) = Ncl(A) and hence K(K(A) = K(A). A⊂NclA, B⊂NclB hence A∪B⊂NclA∪NclB. Now NclA and NclB are net closed sets and hence NclA∪NclB is a net clo ...