# Can a non closed set be compact?

## Can a non closed set be compact?

So a compact set can be open and not closed.

**What is a compact subset?**

A set S⊆R is called compact if every sequence in S has a subsequence that converges to a point in S. One can easily show that closed intervals [a,b] are compact, and compact sets can be thought of as generalizations of such closed bounded intervals. A subset S⊂R is compact iff S is closed and bounded.

**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.

### Is the Cantor set compact?

The Cantor ternary set, and all general Cantor sets, have uncountably many elements, contain no intervals, and are compact, perfect, and nowhere dense.

**Is R N compact?**

R is neither compact nor sequentially compact. That it is not se- quentially compact follows from the fact that R is unbounded and Heine-Borel. To see that it is not compact, simply notice that the open cover consisting exactly of the sets Un = (−n, n) can have no finite subcover.

**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.

## Can a set be closed and unbounded?

Formal definition 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).

**Is a closed subset of a compact set compact?**

37, 2.35] A closed subset of a compact set is compact. Proof : Let K be a compact metric space and F a closed subset. Then its complement Fc is open. Since K is compact, Ω has a finite subcover; removing Fc if necessary, we obtain a finite subcollection of {Vα} which covers F.

**Is R2 compact?**

As the name says itself, the smartphone is compact as it comes in a smaller display size. The phone has a 5.2-inch display size with a resolution of 1080 x 2280 pixels and IGZO type screen with 485 PPI (pixels per inch). Sharp Aquos R2 Compact comes in three different colors, black, white, and green.

### Is hausdorff space compact?

A compact Hausdorff space or compactum, for short, is a topological space which is both a Hausdorff space as well as a compact space. This is precisely the kind of topological space in which every limit of a sequence or more generally of a net that should exist does exist (this prop.) and does so uniquely (this prop).