Can a set be both 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.

Why is null set both open and closed?

In any topological space X, the empty set is open by definition, as is X. Since the complement of an open set is closed and the empty set and X are complements of each other, the empty set is also closed, making it a clopen set. Moreover, the empty set is compact by the fact that every finite set is compact.

Why is R both open and closed?

R is open because any of its points have at least one neighborhood (in fact all) included in it; R is closed because any of its points have every neighborhood having non-empty intersection with R (equivalently punctured neighborhood instead of neighborhood).

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.

Is r2 open and closed?

Example: The blue circle represents the set of points (x, y) satisfying x2 + y2 = r2. The red set is an open set, the blue set is its boundary set, and the union of the red and blue sets is a closed set.

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 the set open closed or neither?

Intuitively, an open set is a set that does not include its “boundary.” Note that not every set is either open or closed, in fact generally most subsets are neither. The set [0,1)⊂R is neither open nor closed.

How should I think of an open vs. closed set?

The red set is an open set, the blue set is its boundary set, and the union of the red and blue sets is a closed set. In mathematics, open sets are a generalization of open intervals in the real line.

What is a set that is both open and closed?

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.

Is the complement of a closed set always open?

A set is a closed set if its complement is open. So is a closed set in since its complement is an open set. Any set with finite cardinality (for example or ) is a closed set. Also observe that the entire set is both a closed and open set with respect to . So a closed or an open set need not be bounded.

What is an example of a closed set?

A set that has closure is not always a closed set. For example, the set of real numbers, for example, has closure when it comes to addition since adding any two real numbers will always give you another real number. However, the set of real numbers is not a closed set as the real numbers can go on to infinity .