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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
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
The concept of open sets can be formalized in various degrees of generality.
A subset U of 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 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.
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.
In topological spaceTopological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studiess, 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.)