open sets

Open set

In metric topology and related fields of mathematics, a set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U. In other words, the distance between any point x in U and the edge of U is always greater than zero.

As an example, consider the open interval consisting of all real numbers x with Here, the topology is the usual topology on the real line. We can look at this in two ways. Since any point in the interval is different from 0 and 1, the distance from that point to the edge is always non-zero. Or equivalently, for any point in the interval we can move by a small enough amount in any direction without touching the edge and still be inside the set. Therefore, the interval is open. However, the interval consisting of all numbers x with is not open in the topology induced from the real line; if one takes x = 1 and moves even the tiniest bit in the positive direction, one will be outside of Taken as a stand-alone space then it is open, since by definition the whole space is both open and closed.

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 interior 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.

Metric spaces

A subset U of a metric space 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 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. 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 Note that infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form where n is a positive integer, is the set {0} which is closed in the real line. Sets that can be constructed as the intersection of countably many open sets are denoted Gδ sets.

The topological definition of open sets generalises the metric space definition: If one begins with a metric space and defines open sets as before, then the family of all open sets is a topology on the metric space. Every metric space is therefore, in a natural way, a topological space. There are, however, topological spaces which are not metric spaces.

Properties

  • The empty set is open (note that the empty set is also closed).
  • The union of any number of open sets is open.
  • The intersection of a finite number of open sets is open.

Uses

Open sets have a fundamental importance in topology. The concept is required to define and make sense for topological space and other topological structures that deal with the notions of closeness and convergence for a space such as metric spaces and uniform spaces.

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.

Note

Note that whether a given set U is open depends on the surrounding space. For instance, if U is defined as the set of rational numbers in the interval then U is open in the rational numbers, but not open in the real numbers. This is because when U is in the rational numbers there are no irrational numbers that can be moved to—the smallest possible displacement is from one rational number to another. Also, no matter how close an element of U is to 0 or 1, there is always another rational number between it and 0 or 1, so from any element of U there is always a way to make a small enough displacement that you can get closer to 0 or 1 while staying inside U. But, when this set is in the real numbers, there are irrational numbers between all of the rational numbers and it is possible to move from an element of U to an irrational number (which is not an element of U). So, for any displacement from some beginning element of U to some ending element, there is always a smaller distance from the beginning element to an irrational number which is outside of U. (Even though the irrational number may be between 0 and 1, it is not in U because U contains only rational numbers.)

Some sets 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. While others are neither open nor closed, such as in R. In fact, the set is the union of the sets and {1}, an open set and a closed set respectively. An important point is that an open set is not the opposite of "closed set", rather a closed set is the complement of an open set.

See also

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