What is the closure of an unbounded set?

What you are looking for. I.e:, Any complement of any open ball! A simple example of a closed but unbounded set is [0,∞).

Can a set be bounded and unbounded?

A function can be bounded at one end, and unbounded at another. A number m ∈ R is the infimum or greatest lower bound of A, if 1.

Can a set be closed but not compact?

Intuitively, a compact set S of R can not go off to infinity (bounded), nor can it accumulate to some point not in S (closed). It must be both bounded and closed. Theorem 2: Let f : X → Y be a continuous map between topological spaces. If S ⊆ X is compact, then f(S) is compact.

Are set sets bounded closed?

A closed set is a bounded set that contains its boundary. A bounded set need not contain its boundary. If it contains none of its boundary, it is open. If it contains all of its boundary, it is closed.

What is an unbounded set?

A set which is bounded above and bounded below is called bounded. A set which is not bounded is called unbounded. For example the interval (−2,3) is bounded. Examples of unbounded sets: (−2,+∞),(−∞,3), the set of all real num- bers (−∞,+∞), the set of all natural numbers.

Can an unbounded set be compact?

We cannot take a finite subcover to cover A. A similar proof shows that an unbounded set is not compact. Continuous images of compact sets are compact.

What does it mean for a set to be unbounded?

How do you prove a set is unbounded?

For every n∈N we have n2≥n, which follows by multiplying both sides of the inequality n≥1 with n and concluding n2≥n≥1. Given any x∈R choose a natural number n>x. Then n2∈A and n2≥n>x. Thus A is unbounded.

Is a closed interval compact?

The bounded closed interval [0, 1] is compact and its maximum 1 and minimum 0 belong to the set, while the open interval (0, 1) is not compact and its supremum 1 and infimum 0 do not belong to the set. The unbounded, closed interval [0, ∞) is not compact, and it has no maximum.

What is the difference between bounded and unbounded?

Bounded and Unbounded Intervals An interval is said to be bounded if both of its endpoints are real numbers. Bounded intervals are also commonly known as finite intervals. Conversely, if neither endpoint is a real number, the interval is said to be unbounded.

Does bounded imply closed?

Clearly bounded does not imply closed.

How do you prove a set is closed?

A set is closed if it contains all its limit points. Proof. Suppose A is closed. Then, by definition, the complement C(A) = X \A is open.