Open sets are the fundamental building blocks of topology. In the familiar setting of a metric space, the open sets have a natural description, which can be thought of as a generalization of an open interval on the real number line. Intuitively, an open set is a set that does not contain its boundary, in the same way that the endpoints of an interval are not contained in the interval.
The standard definition of continuity can be restated quite concisely in terms of open sets, and the elegance of this restatement leads to a powerful generalization of this idea to general topological spaces. In the same way, many other definitions of topological concepts are formulated in general in terms of open sets.
The complement of an open set is a closed set. Many topological properties related to open sets can be restated in terms of closed sets as well.
In all but the last section of this wiki, the setting will be a general metric space Those readers who are not completely comfortable with abstract metric spaces may think of as being where or for concreteness, and the distance function as being the standard Euclidean distance between two points.
An open set in a metric space is a subset of with the following property: for any there is a real number such that any point in that is a distance from is also contained in
For any point define to be the open ball of radius which is the set of all points in that are within a distance from (For instance, if then is the open interval If is the open disk centered at with radius Then a set is open if and only if for each point there is an such that is completely contained in
So the intuition is that an open set is a set for which any point in the set has a small "halo" around it that is completely contained in the set. The idea is that this halo fails to exist precisely when the point lies on the boundary of the set, so the condition that is open is the same as saying that it doesn't contain any of its boundary points. With the correct definition of boundary, this intuition becomes a theorem.
The boundary of a set inside a metric space is the set of points such that for any contains at least one point in and at least one point not in
A subset of a metric space is open if and only if it does not contain any of its boundary points.
It is clear that an open set cannot contain any of its boundary points since the halo condition would not apply to those points. On the other hand, if a set doesn't contain any of its boundary points, that is enough to show that it is open: for every point since is not a boundary point, that implies that there is some ball around that is either contained in or contained in the complement of But every ball around contains at least one point in namely itself, so it must be the former, and has a halo inside
Trivial open sets: The empty set and the entire set are both open. This is a straightforward consequence of the definition.
To see the first statement, consider the halo around a point in the union. Since any in the union is in one of the open sets it has a around it contained in so that ball is contained in the union as well.
The second statement is proved in the below exercise.
Let be a collection of open sets in If is finite, then the intersection is also an open set. Here is a proof:
Suppose For each let be a ball of some positive radius around which is contained entirely inside Then the intersection of the is a ball around which is contained entirely inside the intersection, so the intersection is open.
Here a ball around is a set ( a positive real number) consisting of all points such that In it is an open disk centered at of radius
Where does this proof go wrong when is infinite?
- Basic open sets: Every open set is the union of open balls
To see this, let be an open set and, for each let be the halo around Then each is contained in so their union is; but their union must be all of since every point is contained in (at least) one of them.
This notion of building up open sets by taking unions of certain types of open sets generalizes to abstract topology, where the building blocks are called basic open sets, or a base.
The definition of continuous functions, which includes the epsilon-delta definition of a limit, can be restated in terms of open sets. This reformulation turns out to be the way to generalize the concept of continuity to abstract topological spaces.
Recall that a function is said to be continuous if
A function is continuous if and only if the inverse image of any open set is open. That is, if is an open subset of then is an open subset of
Suppose is continuous, is open, and Then so there is an open ball for some Now since there must exist some such that whenever That is, for all lies in But this ball is contained in so for all So This shows that is open, since we have found a ball around any point which is contained in
The proof of the opposite ("if") direction is similar.
Note that the image of an open set under a continuous function is not necessarily open. For instance, defined by satisfies
The theorem above motivates the general definition of topological continuity: a continuous function between two metric spaces (or topological spaces) is defined to be a function with the property that the inverse image of an open set is open.
The interior of a set is defined to be the largest open subset of It equals the union of every open subset of The interior of is the set of points in which are not boundary points of
A connected set is defined to be a set which is not the disjoint union of two nonempty open sets.
A compact subset of is a subset with the property that every covering of by a collection of open sets has a finite subcover--that is, given a collection of open sets whose union contains it is possible to choose a subcollection of finitely many open sets from the covering whose union still contains
In the absence of a metric, it is possible to recover many of the definitions and properties of metric spaces for arbitrary sets. The idea is, given a set to specify a collection of open subsets (called a topology) satisfying the following axioms:
The empty set and are open.
An infinite union of open sets is open; a finite intersection of open sets is open.
These are, in a sense, the fundamental properties of open sets. These axioms allow for broad generalizations of open sets to contexts in which there is no natural metric. Indeed, there are some important examples of topologies in mathematics which do not come from metrics, including the Zariski topology in algebraic geometry.
- Pred, . Open set - example. Retrieved January 24, 2009, from https://commons.wikimedia.org/wiki/File:Open_set_-_example.png