examples of compact spaces
Here are some examples of compact spaces (http://planetmath.org/Compact^{}):

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The unit interval [0,1] is compact. This follows from the HeineBorel Theorem. Proving that theorem is about as hard as proving directly that [0,1] is compact. The halfopen interval (0,1] is not compact: the open cover $(1/n,1]$ for $n=1,2,\mathrm{\dots}$ does not have a finite subcover.

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Again from the HeineBorel Theorem, we see that the closed unit ball^{} of any finitedimensional normed vector space^{} is compact. This is not true for infinite dimensions^{}; in fact, a normed vector space is finitedimensional if and only if its closed unit ball is compact.

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Any finite topological space is compact.

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Consider the set ${2}^{\mathbb{N}}$ of all infinite sequences with entries in $\{0,1\}$. We can turn it into a metric space by defining $d(({x}_{n}),({y}_{n}))=1/k$, where $k$ is the smallest index such that ${x}_{k}\ne {y}_{k}$ (if there is no such index, then the two sequences are the same, and we define their distance to be zero). Then ${2}^{\mathbb{N}}$ is a compact space, a consequence of Tychonoff^{}’s theorem. In fact, ${2}^{\mathbb{N}}$ is homeomorphic^{} to the Cantor set (which is compact by HeineBorel). This construction can be performed for any finite set, not just {0,1}.

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Consider the set $K$ of all functions $f:\mathbb{R}\to [0,1]$ and defined a topology^{} on $K$ so that a sequence $({f}_{n})$ in $K$ converges towards $f\in K$ if and only if $({f}_{n}(x))$ converges towards $f(x)$ for all $x\in \mathbb{R}$. (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.

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Take any set $X$, and define the cofinite topology^{} 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.

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The prime spectrum of any commutative ring with the Zariski topology^{} is a compact space important in algebraic geometry^{}. These prime spectra are almost never Hausdorff spaces.

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If $H$ is a Hilbert space^{} and $A:H\to H$ is a continuous linear operator, then the spectrum of $A$ is a compact subset of $\u2102$. If $H$ is infinitedimensional, then any compact subset of $\u2102$ arises in this manner from some continuous linear operator $A$ on $H$.

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If $\mathcal{A}$ is a complex C*algebra^{} which is commutative^{} and contains a one, then the set $X$ of all nonzero algebra homomorphisms $\varphi :\mathcal{A}\to \u2102$ carries a natural topology (the weak* topology) which turns it into a compact Hausdorff space. $\mathcal{A}$ is isomorphic^{} to the C*algebra of continuous complexvalued functions on $X$ with the supremum norm.

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

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Any locally compact Hausdorff space^{} can be turned into a compact space by adding a single point to it (Alexandroff onepoint compactification (http://planetmath.org/AlexandrovOnePointCompactification)). The onepoint compactification of $\mathbb{R}$ is homeomorphic to the circle ${S}^{1}$; the onepoint compactification of ${\mathbb{R}}^{2}$ is homeomorphic to the sphere ${S}^{2}$. Using the onepoint compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a nonHausdorff space.

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Other nonHausdorff compact spaces are given by the left order topology (or right order topology) on bounded^{} totally ordered sets^{}.
Title  examples of compact spaces 

Canonical name  ExamplesOfCompactSpaces 
Date of creation  20130322 12:48:47 
Last modified on  20130322 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 