Metric geometry of locally compact groups
... – And the tightly interwoven developments of combinatorial group theory and low dimensional topology, from Dehn to Thurston, and so many others. From 1980 onwards, for all these reasons and under guidance of Gromov, in particular of his articles [Grom–81b, Grom–84, Grom–87, Grom–93], the group commu ...
... – And the tightly interwoven developments of combinatorial group theory and low dimensional topology, from Dehn to Thurston, and so many others. From 1980 onwards, for all these reasons and under guidance of Gromov, in particular of his articles [Grom–81b, Grom–84, Grom–87, Grom–93], the group commu ...
Structure theory of manifolds
... space X together with a maximal P L atlas on X. See Hudson [2] for details. Equivalently a P L structure on X may be defined as a P L equivalence class of triagulations t : |K| → X where K is a countable, locally finite simplicial complex. If X is an open subset of Rn , then X has a standard P L str ...
... space X together with a maximal P L atlas on X. See Hudson [2] for details. Equivalently a P L structure on X may be defined as a P L equivalence class of triagulations t : |K| → X where K is a countable, locally finite simplicial complex. If X is an open subset of Rn , then X has a standard P L str ...
Posets and homotopy
... Homotopy types of infinite posets For infinite P, Q there is in general no known simple way to describe homotopy classes of maps P → Q. One result in this direction is the following (K. 2009): Theorem: Let {fα : P → Q}α≤γ , where γ is a countable ordinal, be a family of continuous maps such that: 1 ...
... Homotopy types of infinite posets For infinite P, Q there is in general no known simple way to describe homotopy classes of maps P → Q. One result in this direction is the following (K. 2009): Theorem: Let {fα : P → Q}α≤γ , where γ is a countable ordinal, be a family of continuous maps such that: 1 ...
Volume 11, 2007 1 MAIN ARTICLES SPACES WITH A LOCALLY
... Lindelöf subset of X for some neighborhood U of y in Y ). (3) f is called a 1-sequence-covering mapping ([23]) if for each y ∈ Y there is x ∈ f −1 (y), such that whenever {yn } is a sequence converging to y in Y , there is a sequence {xn } converging to x in X with each xn ∈ f −1 (yn ). (4) f is ca ...
... Lindelöf subset of X for some neighborhood U of y in Y ). (3) f is called a 1-sequence-covering mapping ([23]) if for each y ∈ Y there is x ∈ f −1 (y), such that whenever {yn } is a sequence converging to y in Y , there is a sequence {xn } converging to x in X with each xn ∈ f −1 (yn ). (4) f is ca ...
Introduction to Combinatorial Homotopy Theory
... required coherence properties, to define the whole collection of morphisms {Xα }α . Let us consider the simplest simplicial complex X, the realization of which is (homeomorphic to) a circle S 1 = {(x, y) ∈ R2 st x2 + y 2 = 1}. Three vertices and three edges are necessary: X = (V, S) with V = 2 = {0, ...
... required coherence properties, to define the whole collection of morphisms {Xα }α . Let us consider the simplest simplicial complex X, the realization of which is (homeomorphic to) a circle S 1 = {(x, y) ∈ R2 st x2 + y 2 = 1}. Three vertices and three edges are necessary: X = (V, S) with V = 2 = {0, ...
Decomposition of Generalized Closed Sets in Supra Topological
... difference of two closed subsets of an n-dimensional Euclidean space. Implicit in their work is the notion of a locally closed subset of a topological space. Bourbaki [2] defined a subset of space (X, τ) is called locally closed, if it is the intersection of an open set and a closed set. Stone [11] ...
... difference of two closed subsets of an n-dimensional Euclidean space. Implicit in their work is the notion of a locally closed subset of a topological space. Bourbaki [2] defined a subset of space (X, τ) is called locally closed, if it is the intersection of an open set and a closed set. Stone [11] ...
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.