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 can't 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. 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 wiggle a tiny 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". 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**. ## Definitions
The concept of open sets can be formalized in various degrees of generality.
### Euclidean space A subset *U* of Euclidean *n*-space **R**^{n} is called *open* if, given any point *x* in *U*, there exists a real number ε > 0 such that, given any point *y* in **R**^{n} 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 **R**^{2} (on a plane) would be all the points within a circle radius *r*, which satisfy the equation . Because the distance of any point *p* in this set from the edge of the set is greater that zero: , 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 generalizes 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**). This 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. |