NationMaster - Encyclopedia: Closeness (topology)

In topology and related areas in mathematics closeness is one of the basic concepts in a topological space. Intuitively we say two sets are close if they are arbitrarily near to each other. The concept can be defined naturally in a metric space where a notion of distance between elements of the space is defined, but it can be generalized to topological spaces where we have no concrete way to measure distances. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

The closure operator closes a given set by mapping it to a closed set which contains the original set and all points close to it. The concept of closeness is related to limit point. In mathematics, given a partially ordered set (P, ≤), a closure operator on P is a function C : P → P with the following properties: if x ≤ y, then C(x) ≤ C(y), i. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ...

1 Definition
2 Properties
3 Closeness relation between a point and a set
4 Closeness relation between two sets
5 Generalized definition
6 See also

Definition

Given a metric space $(X, d)$ we call a point $p$ close to a set $A$ if In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

d (p, A) = 0

Similarly a set $B$ is called close to a set $A$ if

d (B, A) = 0

Properties

if a point $p$ is close to a set $A$ and a set $B$ then $A$ and $B$ are close (the converse is not true!).
closeness between a point and a set is preserved by continuous functions
closeness between two sets is preserved by uniformly continuous functions

In logic, if S is a statement of the form P implies Q, then the converse of S is a statement of the form Q implies P. In general, the verity of S says nothing about the verity of its converse. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

Closeness relation between a point and a set

Let $A$ and $B$ be two sets and $p$ a point.

if $p$ is close to $A$ then $A neq emptyset$
if $p$ is close to $A$ and $B supset A$ then $p$ is close to $B$
if $p$ is close to $A cup B$ then either $p$ is close to $A$ or $p$ is close to $B$

Closeness relation between two sets

Let $A$ , $B$ and $C$ be sets.

if $A$ and $B$ are close then $A neq emptyset$ and $B neq emptyset$
if $A$ and $B$ are close then $B$ and $A$ are close
if $A$ and $B$ are close and $B subset C$ then $A$ and $C$ are close
if $A$ and $B cup C$ are close then either $A$ and $B$ are close or $A$ and $C$ are close
if $A cap B neq emptyset$ then $A$ and $B$ are close

Generalized definition

The closeness relation between a set and a point can be generalized to any topological space. Given a topological space and a point $p$ , $p$ is called close to a set $A$ if $p in cl(A)$ .

To define a closeness relation between two sets the topological structure is too weak and we have to use a uniform structure. Given a uniform space and two sets are called close to each other if they are contained in an entourage. In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence. ... In the mathematical field of topology, a uniform space is a set with a uniform structure. ... In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence. ...

Contents

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Properties

Closeness relation between a point and a set

Closeness relation between two sets

Generalized definition

See also

Definition