an example of a bounded-in-probability, but non-tight
... (3) X is a locally compact separable metric space. In analogy with Definition 1, for a family M of probability measures on X we have: Definition 2 M is tight if from every > 0 there is a compact set K = K() in X such that µ(K) ≥ 1 − for all µ in M . For the Markov chain Φ, the notion of tightne ...
... (3) X is a locally compact separable metric space. In analogy with Definition 1, for a family M of probability measures on X we have: Definition 2 M is tight if from every > 0 there is a compact set K = K() in X such that µ(K) ≥ 1 − for all µ in M . For the Markov chain Φ, the notion of tightne ...
normal and I g - Italian Journal of Pure and Applied Mathematics
... open [18] if A=int(cl(A)) and A is said to be regular closed [18] if A=cl(int(A)). A subset A of a space (X, τ ) is said to be an α-open [11] (resp. preopen [8]) if A⊆int(cl(int(A))) (resp. A⊆int(cl(A))). The complement of α-open set is α-closed [9]. The α-closure [9] of a subset A of X, denoted by ...
... open [18] if A=int(cl(A)) and A is said to be regular closed [18] if A=cl(int(A)). A subset A of a space (X, τ ) is said to be an α-open [11] (resp. preopen [8]) if A⊆int(cl(int(A))) (resp. A⊆int(cl(A))). The complement of α-open set is α-closed [9]. The α-closure [9] of a subset A of X, denoted by ...
Bounded subsets of topological vector spaces
... In this section we are going to study bounded sets in some of the special classes of t.v.s. which we have encountered so far. First of all, let us notice that any ball in a normed space is a bounded set and thus that in normed spaces there exist sets which are at the same time bounded and neighborho ...
... In this section we are going to study bounded sets in some of the special classes of t.v.s. which we have encountered so far. First of all, let us notice that any ball in a normed space is a bounded set and thus that in normed spaces there exist sets which are at the same time bounded and neighborho ...
Spaces not distinguishing convergences of real
... compactness [15] in terms of quasi-uniform or simply-uniform convergence can serve as examples. The referee has informed us about two other sources which show the importance of such study: [19], where Kliś considers normed spaces in which every sequence converging to 0 has a summable subsequence; a ...
... compactness [15] in terms of quasi-uniform or simply-uniform convergence can serve as examples. The referee has informed us about two other sources which show the importance of such study: [19], where Kliś considers normed spaces in which every sequence converging to 0 has a summable subsequence; a ...
Boundaries of CAT(0) Groups and Spaces
... In this case we write γ ∼ γ 0 and we define ∂X := {γ geodesic ray in X}/ ∼ . We call ∂X the boundary of X. In the following we will explore the different topologies that we can use on ∂X. As this definition of the boundary is a definition at large scale, it holds also for δ-hyperbolic spaces. Visual ...
... In this case we write γ ∼ γ 0 and we define ∂X := {γ geodesic ray in X}/ ∼ . We call ∂X the boundary of X. In the following we will explore the different topologies that we can use on ∂X. As this definition of the boundary is a definition at large scale, it holds also for δ-hyperbolic spaces. Visual ...
Note on new Classes of Separation axioms
... Definition 2.1. [9] A subset A of X is called δ-closed in a topological space (X, τ ) if A = δcl(A), where δcl(A) = {x ∈ X : int(cl(U )) ∩ A = ∅, U ∈ O(X, x)}. The complement of δ-closed set in (X, τ ) is called δ-open set in (X, τ ). From [1], lemma 3, δcl(A) = ∩{F ∈ δC(X) : A ⊆ F } and from corol ...
... Definition 2.1. [9] A subset A of X is called δ-closed in a topological space (X, τ ) if A = δcl(A), where δcl(A) = {x ∈ X : int(cl(U )) ∩ A = ∅, U ∈ O(X, x)}. The complement of δ-closed set in (X, τ ) is called δ-open set in (X, τ ). From [1], lemma 3, δcl(A) = ∩{F ∈ δC(X) : A ⊆ F } and from corol ...
Nagata-Smirnov Metrization Theorem.nb
... compact and C is a closed subset of X, C is also compact. Compactness is preserved by continuous functions, so f(C) is also compact. Furthermore, in a Hausdorff space compact sets are closed; thus f(C) is closed, and f-1 is continuous. · Lemma 1.4: Suppose X is a topological space with topology an ...
... compact and C is a closed subset of X, C is also compact. Compactness is preserved by continuous functions, so f(C) is also compact. Furthermore, in a Hausdorff space compact sets are closed; thus f(C) is closed, and f-1 is continuous. · Lemma 1.4: Suppose X is a topological space with topology an ...
INTERSECTION OF SETS WITH n
... intersection of convex sets is nonempty, then their union is starshaped. Hence, the following result of Breen [2] is a generalization of Helly’s theorem for closed sets: A finite family of closed convex sets in Rn has a nonempty intersection if (and only if ) the union of every n + 1 or fewer member ...
... intersection of convex sets is nonempty, then their union is starshaped. Hence, the following result of Breen [2] is a generalization of Helly’s theorem for closed sets: A finite family of closed convex sets in Rn has a nonempty intersection if (and only if ) the union of every n + 1 or fewer member ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.