Compactness (real numbers)

A subset of the set of real numbers (or more generally of Euclidean space ) is compact if and only if it is bounded and closed . ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

It must not contain a sequence that converges but whose limit does not belong to the set. Consequences whose value “ grows beyond all limits ” (ie have no limit value) must not be included.

This article covers a simplified version of compactness as it is in or correct in. The above definition is incorrect in the case of general topological spaces ; the general term is presented in the article “ compact room ”. ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

Equivalent formulations

On the basis of this definition it can be shown: A subset of the real numbers is compact if and only if

if every sequence from the set has a convergent subsequence whose limit value belongs to the subset (this condition defines sequence compactness ), or

if a finite partial cover can be selected from every open cover (this defines cover compactness ).

Generalizations

The concept of compactness can easily be generalized to and to other finite-dimensional vector spaces. ${\ displaystyle \ mathbb {R} ^ {n}}$

New points of view arise with infinite-dimensional spaces and with general topological spaces, see compact space . The connection to the special case is then made using Heine-Borel's theorem . Sequence compactness and coverage compactness may no longer be the same in any topological space.

general definition

Let be a subset of a topological space. is called compact if for every open cover , a finite sub-cover there. That means there is a finite subset and . ${\ displaystyle M}$ ${\ displaystyle M}$ ${\ displaystyle (T_ {i}) _ {i \ in I}}$ ${\ displaystyle M \ subset \ cup _ {i \ in I} T_ {i}}$ ${\ displaystyle M}$ ${\ displaystyle J \ subset I}$ ${\ displaystyle M \ subset \ cup _ {i \ in J} T_ {i}}$

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By definition, these must be open sets, and the property must be demonstrated for each such cover. It is not enough to prove that finite partial covers exist for certain coverages only. ${\ displaystyle T_ {i}}$

Examples

Be and real numbers and . ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle a <b}$

A closed interval is compact. Each convergent sequence in this interval must converge to an interval value. ${\ displaystyle [a, b]}$
The half-open intervals and the open interval are not compact because they are not closed. There are sequences that converge towards an edge point of the interval. ${\ displaystyle] a, b], [a, b [}$ ${\ displaystyle] a, b [}$
The set of real numbers is not compact because it is closed but not bounded. It therefore contains series of numbers, each of which “grows beyond all limits” (for example the set of natural numbers).