Chapter VII. Covering Spaces and Calculation of Fundamental Groups
... the covering is n-sheeted , and we talk about an n-fold covering. Of course, unless the covering is trivial, it is impossible to distinguish the sheets of it, but this does not prevent us from speaking about the number of sheets. On the other hand, we adopt the following agreement. By definition, th ...
... the covering is n-sheeted , and we talk about an n-fold covering. Of course, unless the covering is trivial, it is impossible to distinguish the sheets of it, but this does not prevent us from speaking about the number of sheets. On the other hand, we adopt the following agreement. By definition, th ...
Neighbourhood Systems - PMF-a
... c ∈ F such that c ≤ a, c ≤ b hold. F ⊆ L is said to be a filter on L if F is filtered upper set. The set of all filters on L which don’t contain the bottom element 0 of L is denoted by Fil(L). Definition 3.1. Let X be a set and L be a locale. Let N : X → Fil(L) be a map which assigns to every point ...
... c ∈ F such that c ≤ a, c ≤ b hold. F ⊆ L is said to be a filter on L if F is filtered upper set. The set of all filters on L which don’t contain the bottom element 0 of L is denoted by Fil(L). Definition 3.1. Let X be a set and L be a locale. Let N : X → Fil(L) be a map which assigns to every point ...
Basic categorial constructions 1. Categories and functors
... comparison to description by construction. Often, an item is already uniquely determined by a subset of its desired properties. Often, mapping-theoretic descriptions determine further properties an object must have, without explicit details of its construction. Indeed, the common impulse to overtly ...
... comparison to description by construction. Often, an item is already uniquely determined by a subset of its desired properties. Often, mapping-theoretic descriptions determine further properties an object must have, without explicit details of its construction. Indeed, the common impulse to overtly ...
GAUSS WORDS AND THE TOPOLOGY OF MAP GERMS FROM R3
... If ϕ, ψ are homeomorphisms instead of diffeomorphisms, then we say that f, g are topologically equivalent. We say that f : (R3 , 0) → (R3 , 0) is k-determined if any map germ g with the same k-jet is A-equivalent to f . We say that f is finitely determined if it is k-determined for some k. Let f : U → ...
... If ϕ, ψ are homeomorphisms instead of diffeomorphisms, then we say that f, g are topologically equivalent. We say that f : (R3 , 0) → (R3 , 0) is k-determined if any map germ g with the same k-jet is A-equivalent to f . We say that f is finitely determined if it is k-determined for some k. Let f : U → ...
A -sets and Decompositions of â-A -continuity
... / then z = {X\G : G∈I} is a filter [8]. Given a topological space (X, τ ) with an ideal I on X and if ℘(X) is the set of all subsets of X, a set operator (.)? : ℘(X)→℘(X), called a local function [9] of A with respect to τ and I is defined as follows: for A⊆X, A? (I,τ )={x∈X | U∩A∈I / for every U∈τ ...
... / then z = {X\G : G∈I} is a filter [8]. Given a topological space (X, τ ) with an ideal I on X and if ℘(X) is the set of all subsets of X, a set operator (.)? : ℘(X)→℘(X), called a local function [9] of A with respect to τ and I is defined as follows: for A⊆X, A? (I,τ )={x∈X | U∩A∈I / for every U∈τ ...
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.