- "In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space."
A topological space is compact if every open cover of it has a finite subcover. It's a fundamental property that is used in many areas of mathematics.
Metric spaces: A metric space is a set with a distance function defined between its points. It is a natural starting point for understanding compactness.
Topological spaces: A topological space is a set with a topology defined on it, which gives a sense of continuity and convergence. Compactness is a property of topological spaces.
Open sets and closed sets: A set can be open, meaning that all points within it are interior points, or closed, meaning that it contains all its boundary points. A set is compact if every open cover has a finite subcover.
Compactness definitions: There are several equivalent definitions of compactness, including the Heine-Borel theorem, sequential compactness, and limit point compactness.
Properties of compact spaces: Compact spaces have many important properties, such as being closed under continuous maps, being sequentially compact, and being a limit of a convergent sequence of points.
Examples of compact spaces: Examples include the closed interval [0,1], the unit sphere S^n, and the Cantor set. Understanding these examples can help develop an intuition for what it means to be compact.
Compactness and completeness: There is a close relationship between compactness and completeness, which is the property of having all Cauchy sequences converge. Many important theorems in topology involve both compactness and completeness.
Applications of compactness: Compactness is a fundamental concept in topology, with many applications in other areas of mathematics, such as integration theory and differential equations.
Non-compact spaces: It is also important to understand non-compact spaces, which do not have the finite subcover property. Examples include the real line and the plane.
Sequential compactness: A space is sequentially compact if every sequence of points has a converging subsequence. This is a weaker form of compactness than the standard definition, as it only requires "sequential" convergence, but not necessarily "topological" convergence. For example, any closed and bounded subset of Euclidean space is sequentially compact by the Bolzano-Weierstrass theorem, but not all such sets are compact in the usual sense.
Countable compactness: A space is countably compact if every countable open cover has a finite subcover. This is also weaker than the standard definition of compactness, but still useful in some contexts. For example, the first uncountable ordinal with the order topology is not compact, but is countably compact.
Finite compactness: A space is finitely compact if every finite open cover has a finite subcover. This is another weakening of compactness, and corresponds to the idea of a space being "finite" in some sense, or having only "finite" many distinguishable open sets. For example, any discrete space is finitely compact.
Hausdorff compactness: A space is Hausdorff compact if it is compact and Hausdorff. The Hausdorff condition requires that every pair of distinct points can be separated by disjoint open sets, which ensures some nice separation properties. For example, any compact metric space is Hausdorff compact, and the Tychonoff theorem states that the product of any family of Hausdorff compact spaces is again Hausdorff compact.
Algebraic compactness: A space is algebraically compact if it is the spectrum of a commutative unital ring that is Noetherian, which means every ascending chain of ideals eventually stabilizes. This is a more abstract version of compactness that relates a topological space to the structure of its algebraic functions. For example, the affine line over any algebraically closed field is algebraically compact.
Coherent compactness: A space is coherent compact if it is the spectrum of a coherent sheaf of rings on a scheme. This is another algebraic version of compactness that relates a topological space to the coherence of its "sheafy" functions. For example, any proper scheme over a field is coherent compact.
Etale compactness: A space is etale compact if it is the etale site of a quasi-compact and quasi-separated scheme. This is a more sophisticated version of compactness that uses the language of algebraic geometry and sheaf theory. For example, any smooth Deligne-Mumford stack of finite type over a field is etale compact.
- "The idea is that a compact space has no 'punctures' or 'missing endpoints', i.e., it includes all limiting values of points."
- "For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1."
- "The closed interval [0,1] would be compact."
- "The space of rational numbers (Q) is not compact because it has infinitely many 'punctures' corresponding to the irrational numbers."
- "The space of real numbers (R) is not compact either because it excludes the two limiting values +∞ and -∞."
- "However, the extended real number line would be compact since it contains both infinities."
- "These ways usually agree in a metric space, but may not be equivalent in other topological spaces."
- "One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space."
- "The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded."
- "Thus, if one chooses an infinite number of points in the closed unit interval [0, 1], some of those points will get arbitrarily close to some real number in that space."
- "Since neither 0 nor 1 are members of the open unit interval (0, 1), those same sets of points would not accumulate to any point of it, so the open unit interval is not compact."
- "Although subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact."
- "For example, considering R1 (the real number line), the sequence of points 0, 1, 2, 3,... has no subsequence that converges to any real number."
- "Compactness was formally introduced by Maurice Fréchet in 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points to spaces of functions."
- "The Arzelà–Ascoli theorem and the Peano existence theorem exemplify applications of this notion of compactness to classical analysis."
- "Following its initial introduction, various equivalent notions of compactness, including sequential compactness and limit point compactness, were developed in general metric spaces."
- "In general topological spaces, however, these notions of compactness are not necessarily equivalent."
- "The most useful notion — and the standard definition of the unqualified term compactness — is phrased in terms of the existence of finite families of open sets that 'cover' the space."
- "This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets."