## Is compact set closed and bounded?

A subset of Euclidean space in particular is called compact if it is closed and bounded. This implies, by the Bolzano–Weierstrass theorem, that any infinite sequence from the set has a subsequence that converges to a point in the set.

## Are compact sets always bounded?

Compact sets in metric spaces are always bounded. Let k∈K be an arbitrary point, then the sequence of open balls {x∣d(k,x)

Can a non closed set be compact?

No. A compact set need not be closed. Consider any set Y with trivial topology i.e. only open sets are Y and empty set.

Are bounded sets closed?

The whole space is closed, certainly not bounded. The set of points with integer coordinates is closed, not bounded.

### Are unbounded sets closed?

If a set is both closed and unbounded, then it is a club set. Closed proper classes are also of interest (every proper class of ordinals is unbounded in the class of all ordinals). . In fact a club set is nothing else but the range of a normal function (i.e. increasing and continuous).

### Are all compact sets closed?

every compact set is closed, but not conversely. There are, however, spaces in which the compact sets coincide with the closed sets-compact Hausdorff spaces, for example. It is the intent of this note to give several characterizations of such spaces and to list some of their properties.

Is every compact set closed?

Compact sets need not be closed in a general topological space. For example, consider the set {a,b} with the topology {∅,{a},{a,b}} (this is known as the Sierpinski Two-Point Space). The set {a} is compact since it is finite.

Are open set bounded?

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. If it if it contains some but not all of its boundary, it is neither open nor closed.

#### What is bounded and unbounded?

Generally, and by definition, things that are bounded can not be infinite. A bounded anything has to be able to be contained along some parameters. Unbounded means the opposite, that it cannot be contained without having a maximum or minimum of infinity.

Can a compact set be open?

A metric space is a Hausdorff space, so compact sets are closed. Therefore a compact open set must be both open and closed. If X is a connected metric space, then the only candidates are ∅ and X. For example, if X⊂Rn then X is open and compact (in the subspace topology) if and only if X is bounded.