Open set

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In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can "wiggle" or "change" any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U, it cannot be on the edge of U.

As a typical example, consider the open interval (0,1) consisting of all real numbers x with 0 < x < 1. Here, the topology is the usual topology on the real line. If you "wiggle" such an x a little bit (but not too much), then the wiggled version will still be a number between 0 and 1. Therefore, the interval (0,1) is open. However, the interval (0,1] consisting of all numbers x with 0 < x ≤ 1 is not open; if you take x = 1 and move even the tiniest bit in the positive direction, you will be outside of (0,1].

Note that whether a given set U is open depends on the surrounding space, the "wiggle room". For instance, the set of rational numbers between 0 and 1 (exclusive) is open in the rational numbers, but it is not open in the real numbers. Note also that "open" is not the opposite of "closed" (a closed set is the complement of an open set). First, there are sets which are both open and closed (called clopen sets); in R and other connected spaces, only the empty set and the whole space are clopen, while the set of all rational numbers smaller than √2 is clopen in the rationals. Also, there are sets which are neither open nor closed, such as (0,1] in R.

Image:Red blue circle.png

Example: The points satisfying <math>r=\sqrt{x^2+y^2}</math> are colored blue. The points satisfying <math>r<\sqrt{x^2+y^2}</math> are colored red. The red points form an open set. The union of the red and blue points is a closed set.

1 Definitions
- 1.1 Function-analytic
- 1.2 Euclidean space
- 1.3 Metric spaces
- 1.4 Topological spaces
2 Uses
3 Manifolds
4 See also

Definitions

The concept of open sets can be formalized in various degrees of generality.

Function-analytic

A point set in Rⁿ is called open when every point P of the set is an inner point.

Euclidean space

A subset U of the Euclidean n-space Rⁿ is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in Rⁿ whose Euclidean distance from x is smaller than ε, y also belongs to U. (Equivalently, U is open if every point in U has a neighbourhood contained in U)

Intuitively, ε measures the size of the allowed "wiggles".

An example of an open set in R² (on a plane) would be all the points within a circle of radius r, which satisfy the inequality <math>r>\sqrt{x^2+y^2}</math>.

Because the distance of any point p in this set from the edge of the set is greater than zero: <math>r-\sqrt{x^2+y^2}>0</math>, we can set ε to half of this distance, which means ε is also greater than zero, and all the points that are within a distance of ε to p are also in the set, thus satisfying the conditions for an open set.

Metric spaces

A subset U of a metric space (M,d) is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with d(x,y) < ε, y also belongs to U. (Equivalently, U is open if every point in U has a neighbourhood contained in U)

This generalises the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.

Topological spaces

In topological spaces, the concept of openness is taken to be fundamental. One starts with an arbitrary set X and a family of subsets of X satisfying certain properties that every "reasonable" notion of openness is supposed to have. (Specifically: the union of open sets is open, the finite intersection of open sets is open, and in particular the empty set and X itself are open.) Such a family T of subsets is called a topology on X, and the members of the family are called the open sets of the topological space (X,T). Note that infinite intersections of open sets need not be open. Sets that can be constructed as the intersection of countably many open sets are denoted G_δ sets.

The topological definition of open sets generalises the metric space definition: If you start with a metric space and define open sets as before, then the family of all open sets will form a topology on the metric space. Every metric space is hence in a natural way a topological space. (There are however topological spaces which are not metric spaces.)

Uses

Every subset A of a topological space X contains a (possibly empty) open set; the largest such open set is called the interior of A. It can be constructed by taking the union of all the open sets contained in A.

Given topological spaces X and Y, a function f from X to Y is continuous if the preimage of every open set in Y is open in X. The map f is called open if the image of every open set in X is open in Y.

An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals.

Manifolds

A manifold is called open if it is a manifold without boundary and if it is not compact. This notion differs somewhat from the openness discussed above.