APPLICATIONS OF THE TARSKI–KANTOROVITCH FIXED
... Abstract. We show how some results of the theory of iterated function systems can be derived from the Tarski–Kantorovitch fixed–point principle for maps on partially ordered sets. In particular, this principle yields, without using the Hausdorff metric, the Hutchinson–Barnsley theorem with the only ...
... Abstract. We show how some results of the theory of iterated function systems can be derived from the Tarski–Kantorovitch fixed–point principle for maps on partially ordered sets. In particular, this principle yields, without using the Hausdorff metric, the Hutchinson–Barnsley theorem with the only ...
LINE BUNDLES AND DIVISORS IN ALGEBRAIC GEOMETRY
... It has been said that Chapter II, §§6-7 in Hartshorne [3] (Divisors and Projective Morphisms, respectively) are among the most important parts of the book. Unfortunately, the treatment of the subject is inscrutably abstract, especially for someone struggling to understand schemes (like me). In writi ...
... It has been said that Chapter II, §§6-7 in Hartshorne [3] (Divisors and Projective Morphisms, respectively) are among the most important parts of the book. Unfortunately, the treatment of the subject is inscrutably abstract, especially for someone struggling to understand schemes (like me). In writi ...
Math 210B. Artin–Rees and completions 1. Definitions and an
... of open neighborhoods of x is given by those sequences x = (x0n ) such that x0n = xn for all n ≤ N (with growing N ). This is the subspace topology from the product topologies of the discrete spaces M/I n M , so it is Hausdorff. More explicitly, if x 6= x0 then xn0 6= x0n0 for some n0 , so the open ...
... of open neighborhoods of x is given by those sequences x = (x0n ) such that x0n = xn for all n ≤ N (with growing N ). This is the subspace topology from the product topologies of the discrete spaces M/I n M , so it is Hausdorff. More explicitly, if x 6= x0 then xn0 6= x0n0 for some n0 , so the open ...
Handout on bases of topologies
... A family B of subsets of X is called a basis for T or a base for T if (1) For every B ∈ B we have B ∈ T and (2) Every nonempty set U ∈ T can be represented as a union of some family of sets from B. If B is a basis for a topology T on X, we also say that B generates T. Theorem 1 [Criterion for being ...
... A family B of subsets of X is called a basis for T or a base for T if (1) For every B ∈ B we have B ∈ T and (2) Every nonempty set U ∈ T can be represented as a union of some family of sets from B. If B is a basis for a topology T on X, we also say that B generates T. Theorem 1 [Criterion for being ...
§24 Generators and Commutators
... {x1,x2, . . . ,xn} . In particular, if X = {x} consists of a single element, then x = {x} is the cyclic group generated by x, as we introduced in Definition 11.1. Definitions 11.1 and 24.1 are consistent, as will be proved in Lemma 24.2, below. Our notation ...
... {x1,x2, . . . ,xn} . In particular, if X = {x} consists of a single element, then x = {x} is the cyclic group generated by x, as we introduced in Definition 11.1. Definitions 11.1 and 24.1 are consistent, as will be proved in Lemma 24.2, below. Our notation ...
i?-THEORY FOR MARKOV CHAINS ON A TOPOLOGICAL STATE
... (ii) if{X,,} is R-recurrent and aperiodic, every KGX is an L-set if (i) holds, and 08 also contains an element on which f(x) is bounded from zero. Proof Since M is totally finite, any Be0& can be inner approximated by a zero set B' in B so that M(B') > M(B)-s, [7; p. 171]. Further, there exists an e ...
... (ii) if{X,,} is R-recurrent and aperiodic, every KGX is an L-set if (i) holds, and 08 also contains an element on which f(x) is bounded from zero. Proof Since M is totally finite, any Be0& can be inner approximated by a zero set B' in B so that M(B') > M(B)-s, [7; p. 171]. Further, there exists an e ...
Elements of Functional Analysis - University of South Carolina
... Choose x ∈ X with p(x) < 1. We may firstly assume that x 6= 0 so that p(x) > 0 since we know that 0 ∈ C. Then there is a 0 < λ0 < 1 so that x ∈ λ0 C. Define y = λ10 x so that y ∈ C. Since C is convex we have λ0 y + (1 − λ0 ) · 0 = x ∈ C. We now show reverse inclusion. Fix an x ∈ C and define a map ...
... Choose x ∈ X with p(x) < 1. We may firstly assume that x 6= 0 so that p(x) > 0 since we know that 0 ∈ C. Then there is a 0 < λ0 < 1 so that x ∈ λ0 C. Define y = λ10 x so that y ∈ C. Since C is convex we have λ0 y + (1 − λ0 ) · 0 = x ∈ C. We now show reverse inclusion. Fix an x ∈ C and define a map ...
Covering space
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below. In this case, C is called a covering space and X the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a ""sufficiently good"" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.