examples of compact spaces

Here are some examples of compact spaces (http://planetmath.org/CompactPlanetmathPlanetmath):

  • The unit interval [0,1] is compact. This follows from the Heine-Borel Theorem. Proving that theorem is about as hard as proving directly that [0,1] is compact. The half-open interval (0,1] is not compact: the open cover (1/n,1] for n=1,2, does not have a finite subcover.

  • Again from the Heine-Borel Theorem, we see that the closed unit ballPlanetmathPlanetmath of any finite-dimensional normed vector spacePlanetmathPlanetmath is compact. This is not true for infinite dimensionsMathworldPlanetmathPlanetmathPlanetmath; in fact, a normed vector space is finite-dimensional if and only if its closed unit ball is compact.

  • Any finite topological space is compact.

  • Consider the set 2 of all infinite sequences with entries in {0,1}. We can turn it into a metric space by defining d((xn),(yn))=1/k, where k is the smallest index such that xkyk (if there is no such index, then the two sequences are the same, and we define their distance to be zero). Then 2 is a compact space, a consequence of TychonoffPlanetmathPlanetmath’s theorem. In fact, 2 is homeomorphicMathworldPlanetmath to the Cantor set (which is compact by Heine-Borel). This construction can be performed for any finite set, not just {0,1}.

  • Consider the set K of all functions f:[0,1] and defined a topologyMathworldPlanetmath on K so that a sequence (fn) in K converges towards fK if and only if (fn(x)) converges towards f(x) for all x. (There is only one such topology; it is called the topology of pointwise convergence). Then K is a compact topological space, again a consequence of Tychonoff’s theorem.

  • Take any set X, and define the cofinite topologyMathworldPlanetmath on X by declaring a subset of X to be open if and only if it is empty or its complement is finite. Then X is a compact topological space.

  • The prime spectrum of any commutative ring with the Zariski topologyMathworldPlanetmath is a compact space important in algebraic geometryMathworldPlanetmathPlanetmath. These prime spectra are almost never Hausdorff spaces.

  • If H is a Hilbert spaceMathworldPlanetmath and A:HH is a continuous linear operator, then the spectrum of A is a compact subset of . If H is infinite-dimensional, then any compact subset of arises in this manner from some continuous linear operator A on H.

  • If 𝒜 is a complex C*-algebraPlanetmathPlanetmath which is commutativePlanetmathPlanetmathPlanetmath and contains a one, then the set X of all non-zero algebra homomorphisms ϕ:𝒜 carries a natural topology (the weak-* topology) which turns it into a compact Hausdorff space. 𝒜 is isomorphicPlanetmathPlanetmathPlanetmath to the C*-algebra of continuous complex-valued functions on X with the supremum norm.

  • Any profinite group is compact Hausdorff: finite discrete spaces are compact Hausdorff, therefore their productPlanetmathPlanetmath is compact Hausdorff, and a profinite group is a closed subset of such a product.

  • Any locally compact Hausdorff spacePlanetmathPlanetmath can be turned into a compact space by adding a single point to it (Alexandroff one-point compactification (http://planetmath.org/AlexandrovOnePointCompactification)). The one-point compactification of is homeomorphic to the circle S1; the one-point compactification of 2 is homeomorphic to the sphere S2. Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.

  • Other non-Hausdorff compact spaces are given by the left order topology (or right order topology) on boundedPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath totally ordered setsMathworldPlanetmath.

Title examples of compact spaces
Canonical name ExamplesOfCompactSpaces
Date of creation 2013-03-22 12:48:47
Last modified on 2013-03-22 12:48:47
Owner yark (2760)
Last modified by yark (2760)
Numerical id 16
Author yark (2760)
Entry type Example
Classification msc 54D30
Related topic TopologicalSpace