SUBDIVISIONS OF SMALL CATEGORIES Let A be a
... category with a single object, BG is called the classsifying space of the group G. The space BG is often written as K(G, 1) and called an Eilenberg-Mac Lane space. It is characterized (up to homotopy type) as a connected space with π1 (K(G, 1)) = G and with all higher homotopy groups πq (K(G, 1)) = ...
... category with a single object, BG is called the classsifying space of the group G. The space BG is often written as K(G, 1) and called an Eilenberg-Mac Lane space. It is characterized (up to homotopy type) as a connected space with π1 (K(G, 1)) = G and with all higher homotopy groups πq (K(G, 1)) = ...
Subdivide.pdf
... category with a single object, BG is called the classsifying space of the group G. The space BG is often written as K(G, 1) and called an Eilenberg-Mac Lane space. It is characterized (up to homotopy type) as a connected space with π1 (K(G, 1)) = G and with all higher homotopy groups πq (K(G, 1)) = ...
... category with a single object, BG is called the classsifying space of the group G. The space BG is often written as K(G, 1) and called an Eilenberg-Mac Lane space. It is characterized (up to homotopy type) as a connected space with π1 (K(G, 1)) = G and with all higher homotopy groups πq (K(G, 1)) = ...
Equivariant asymptotic dimension, Damian Sawicki, praca magisterska
... When a group G acts on a topological space X (on a set Y ), we will shortly say that X (Y ) is a G-space (a G-set). G\X will denote a quotient of the set (the topological space) X under the left action of G. A similar notation “X \ Y ” will be used to denote the set-theoretical difference X minus Y ...
... When a group G acts on a topological space X (on a set Y ), we will shortly say that X (Y ) is a G-space (a G-set). G\X will denote a quotient of the set (the topological space) X under the left action of G. A similar notation “X \ Y ” will be used to denote the set-theoretical difference X minus Y ...
1. Theorem: If (X,d) is a metric space, then the following are
... 1. Theorem: If (X,d) is a metric space, then the following are equivalent: (a) X is second countable (b) X is Lindelöf. (c) X is separable. Proof: (a) → (b) and (a) → (c) are true for general topological spaces by the tree from our notes. Therefore they are true for metric spaces. (c) → (a): Suppos ...
... 1. Theorem: If (X,d) is a metric space, then the following are equivalent: (a) X is second countable (b) X is Lindelöf. (c) X is separable. Proof: (a) → (b) and (a) → (c) are true for general topological spaces by the tree from our notes. Therefore they are true for metric spaces. (c) → (a): Suppos ...
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.