Is the intersection of an open and closed set closed?

Is the intersection of an open and closed set closed?

The intersection of a finite number of open sets is open. An arbitrary (finite, countable, or uncountable) intersection of closed sets is closed. The union of a finite number of closed sets is closed.

Can a closed set equal the intersection of open sets?

Let’s take a metric space. Then any closed set can be written as a countable intersection of open sets.

Can a set be open and closed at the same time?

Sets can be open, closed, both, or neither. (A set that is both open and closed is sometimes called “clopen.”) The definition of “closed” involves some amount of “opposite-ness,” in that the complement of a set is kind of its “opposite,” but closed and open themselves are not opposites.

What is a set that is both open and closed?

In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open and closed, and therefore clopen.

Is the intersection of open sets open?

The intersection of a finite number of open sets is open. A complement of an open set (relative to the space that the topology is defined on) is called a closed set. A set may be both open and closed (a clopen set).

How do you show a set is open and closed?

To prove that a set is open, one can use one of the following: — Use the definition, that is prove that every point in the set is an interior point. — Prove that its complement is closed. — Prove that it can be written as the intersection of a finite family of open sets or as the union of a family of open sets.

What is open set example?

Definition. An open subset of R is a subset E of R such that for every x in E there exists ϵ > 0 such that Bϵ(x) is contained in E. For example, the open interval (2,5) is an open set. Both R and the empty set are open.

Is Z closed set?

Note that Z is a discrete subset of R. Thus every converging sequence of integers is eventually constant, so the limit must be an integer. This shows that Z contains all of its limit points and is thus closed.

Can the intersection of an infinite number of open sets be closed?

I learned that the union of open sets is always open and the intersection of a finite set of open sets is open. However, the intersection of an infinite number of open sets can be closed. Apparently, the following example illustrates this. In E 2, let X be the infinite family of concentric open disks of radius 1 + 1 / n for all n ∈ Z +.

Is the Union of two open sets always open?

I learned that the union of open sets is always open and the intersection of a finite set of open sets is open. However, the intersection of an infinite number of open sets can be closed.

What is the de nition of a closed set?

In fact, many people actually use this as the de nition of a closed set, and then the de nition we’re using, given above, becomes a theorem that provides a characterization of closed sets as complements of open sets. Theorem: A set is closed if and only if it contains all its limit points.

When is the intersection of a topological space open or closed?

If B ⊂ A, then A ∩ B = B is closed. In the case where the topological space is R endowed with the usual topology, A = ( − 1, 1), and B = [ 0, 2], the intersection is A ∩ B = [ 0, 1), which is neither open nor closed. For most topologies one encounters, this case is typical, at last when the intersection is nonempty.