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Analogously: (1) every [[continuous function]] on a compact space is bounded, (2) every continuous function on a compact space achieves a maximum and minimum value, and (3) in a compact metric space, every sequence has a convergent subsequence. |
Analogously: (1) every [[continuous function]] on a compact space is bounded, (2) every continuous function on a compact space achieves a maximum and minimum value, and (3) in a compact metric space, every sequence has a convergent subsequence. |
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For example, the [[real line]] is not compact, since the sequence of [[natural number]]s has no real limit. The open [[interval (mathematics)|interval]] (0, 1) is not compact because it omits its limit points 0 and 1, whereas the closed interval [0, 1] is compact. Similarly, the space of [[rational number]]s <math>\mathbb{Q}</math> is not compact: sequences of rational numbers may converge in the real line to limits (such as <math>\sqrt{2}</math>) that are not rational. The [[ |
For example, the [[real line]] is not compact, since the sequence of [[natural number]]s has no real limit. The open [[interval (mathematics)|interval]] (0, 1) is not compact because it omits its limit points 0 and 1, whereas the closed interval [0, 1] is compact. Similarly, the space of [[rational number]]s <math>\mathbb{Q}</math> is not compact: sequences of rational numbers may converge in the real line to limits (such as <math>\sqrt{2}</math>) that are not rational. The [[ theorem]] states that a subset of the real line is compact [[if and only if]] it is [[closed set|closed]] and [[bounded set|bounded]]; more generally, this theorem is true in any (finite dimensional) [[Euclidean space]]. The [[extended real numbers|extended real line]] is compact in its natural order topology, because adding ±∞ produces a closed and bounded compact interval. |
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Revision as of 21:02, 21 November 2025
Type of mathematical space
In the mathematical field of topology, compactness is a property that generalizes many of the behaviors of finite sets to spaces that may be infinite. A finite set satisfies several useful properties: (1) every function on a finite set is bounded, (2) every function attains a maximum and minimum value, and (3) every infinite sequence of elements must repeat some value infinitely often (pigeonhole principle).
Analogously: (1) every continuous function on a compact space is bounded, (2) every continuous function on a compact space achieves a maximum and minimum value, and (3) in a compact metric space, every sequence has a convergent subsequence.
For example, the real line is not compact, since the sequence of natural numbers has no real limit. The open interval (0, 1) is not compact because it omits its limit points 0 and 1, whereas the closed interval [0, 1] is compact. Similarly, the space of rational numbers Q {\displaystyle \mathbb {Q} } is not compact: sequences of rational numbers may converge in the real line to limits (such as 2 {\displaystyle {\sqrt {2}}} ) that are not rational. The Bolzano–Weierstrass theorem states that a subset of the real line is compact if and only if it is closed and bounded; more generally, this theorem is true in any (finite dimensional) Euclidean space. The extended real line is compact in its natural order topology, because adding ±∞ produces a closed and bounded compact interval.
