Homework Solutions 2
... the upper end of the interval approaches 1. On the other hand, 0 and 1 belong to all of these intervals. So the interection is [0, 1] . This set is closed, but not open. 2.2a Give M the discrete metric. We will show that this gives the right topology, and thus that in this metric space every subset ...
... the upper end of the interval approaches 1. On the other hand, 0 and 1 belong to all of these intervals. So the interection is [0, 1] . This set is closed, but not open. 2.2a Give M the discrete metric. We will show that this gives the right topology, and thus that in this metric space every subset ...
- International Journal of Mathematics And Its Applications
... the set of all subsets of X, a set operator (.)? : P(X) → P(X), called the local function [5] of A with respect to τ and I, is defined as follows: For A ⊂ X, A? (τ, I) = {x ∈ X|U ∩ A ∈ / I for every open neighbourhood U of x}. A Kuratowski closure operator Cl? (.) for a topology τ ? (τ, I) called th ...
... the set of all subsets of X, a set operator (.)? : P(X) → P(X), called the local function [5] of A with respect to τ and I, is defined as follows: For A ⊂ X, A? (τ, I) = {x ∈ X|U ∩ A ∈ / I for every open neighbourhood U of x}. A Kuratowski closure operator Cl? (.) for a topology τ ? (τ, I) called th ...
The greatest splitting topology and semiregularity
... space X , the continuity of a map g : X Y ! Z implies that of the map gb : X ! Ct(Y; Z ) de ned by relation gb (x)(y ) = g (x; y ) for every x 2 X and y 2 Y . A topology t on C (Y; Z ) is called admissible if for every space X , the continuity of a map f : X ! Ct(Y; Z ) implies that of the map ff ...
... space X , the continuity of a map g : X Y ! Z implies that of the map gb : X ! Ct(Y; Z ) de ned by relation gb (x)(y ) = g (x; y ) for every x 2 X and y 2 Y . A topology t on C (Y; Z ) is called admissible if for every space X , the continuity of a map f : X ! Ct(Y; Z ) implies that of the map ff ...
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.