Baire Spaces and the Wijsman Topology
... separable metrizable space. Then X is Baire if and only if (F(X),Twd) is Baire for each compatible metric d on X. A space is almost locally separable, provided that the set of points of local separability is dense. In a metrizable space, this is equivalent to having a countablein-itself π-base. Co ...
... separable metrizable space. Then X is Baire if and only if (F(X),Twd) is Baire for each compatible metric d on X. A space is almost locally separable, provided that the set of points of local separability is dense. In a metrizable space, this is equivalent to having a countablein-itself π-base. Co ...
On m-Quasi-Irresolute Functions
... called a minimal structure (briefly m-structure) on X if m satisfies the following properties: ∅ ∈ mX and X ∈ mX . By (X, mX ), we denote a nonempty subset X with normal structure mX on X. We call the pair (X, mX ) an m-space. Each member of mX is said to be mX -open (briefly m-open) and the complem ...
... called a minimal structure (briefly m-structure) on X if m satisfies the following properties: ∅ ∈ mX and X ∈ mX . By (X, mX ), we denote a nonempty subset X with normal structure mX on X. We call the pair (X, mX ) an m-space. Each member of mX is said to be mX -open (briefly m-open) and the complem ...
On Separation Axioms and Sequences
... The family of all β-θ-open (resp. β-θ-closed) sets of X containing a point x ∈ X is denoted by βθO(X, x) (resp. βθC(X, x)). The family of all β-θ-open (resp. β-θ-closed) sets in X is denoted by βθO(X) (resp. βθC(X)). 3. Separation axioms and sequences In this section, we introduce and study β-θ-sepa ...
... The family of all β-θ-open (resp. β-θ-closed) sets of X containing a point x ∈ X is denoted by βθO(X, x) (resp. βθC(X, x)). The family of all β-θ-open (resp. β-θ-closed) sets in X is denoted by βθO(X) (resp. βθC(X)). 3. Separation axioms and sequences In this section, we introduce and study β-θ-sepa ...
On topologies defined by irreducible sets
... respect to the specialization order of Alexandroff topology exists). Such a way of defining the Scott topology from the Alexandroff topology leads us to an idea of defining a new topology, called the irreduciblyderived topology, from any given T0 topology on a set. In this paper, we shall investigat ...
... respect to the specialization order of Alexandroff topology exists). Such a way of defining the Scott topology from the Alexandroff topology leads us to an idea of defining a new topology, called the irreduciblyderived topology, from any given T0 topology on a set. In this paper, we shall investigat ...
Lecture 9: Tangential structures We begin with some examples of
... of the tangent bundle. The general definition allows for more exotic possibilities. We move from a geometric description—and an extensive discussion of orientations and spin structures—to a more abstract topological definition. Note there are both stable and unstable tangential structures. The stabl ...
... of the tangent bundle. The general definition allows for more exotic possibilities. We move from a geometric description—and an extensive discussion of orientations and spin structures—to a more abstract topological definition. Note there are both stable and unstable tangential structures. The stabl ...
On some locally closed sets and spaces in Ideal Topological
... (ii) A = Ucl(A) for some δ̂ s - open set U. (iii) cl(A) – A is δ̂ s - closed. (iv) A(X–cl(A)) is δ̂ s - open. Proof. (i)(ii) If A δ̂ sILC, then there exist a δ̂ s – open set U and a -I-closed set F such that A = UF. Clearly AUcl(A). Since F is -I-closed, cl(A) cl(F) = F and so Uc ...
... (ii) A = Ucl(A) for some δ̂ s - open set U. (iii) cl(A) – A is δ̂ s - closed. (iv) A(X–cl(A)) is δ̂ s - open. Proof. (i)(ii) If A δ̂ sILC, then there exist a δ̂ s – open set U and a -I-closed set F such that A = UF. Clearly AUcl(A). Since F is -I-closed, cl(A) cl(F) = F and so Uc ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.