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 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.
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Again from the Heine-Borel Theorem, we see that the closed unit ball of any finite-dimensional normed vector space is compact. This is not true for infinite dimensions; in fact, a normed vector space is finite-dimensional 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 $x_{k}\ot=y_{k}$ (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 Heine-Borel). This construction can be performed for any finite set, not just {0,1}.
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Consider the set $K$ of all functions $f:\\mathbb{R}\\rightarrow[0,1]$ 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 $A:H\\rightarrow H$ is a continuous linear operator, then the spectrum of $A$ is a compact subset of $\\mathbb{C}$ . If $H$ is infinite-dimensional, then any compact subset of $\\mathbb{C}$ arises in this manner from some continuous linear operator $A$ on $H$ .
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If $\\cal{A}$ is a complex C*-algebra which is commutative and contains a one, then the set $X$ of all non-zero algebra homomorphisms $\\phi:\\cal{A}\\rightarrow\\mathbb{C}$ carries a natural topology (the weak-* topology) which turns it into a compact Hausdorff space. $\\cal{A}$ is isomorphic to the C*-algebra of continuous complex-valued 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 one-point compactification (http://planetmath.org/AlexandrovOnePointCompactification)). The one-point compactification of $\\mathbb{R}$ is homeomorphic to the circle $S^{1}$ ; the one-point compactification of $\\mathbb{R}^{2}$ is homeomorphic to the sphere $S^{2}$ . Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.
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Other non-Hausdorff compact spaces are given by the left order topology (or right order topology) on bounded totally ordered sets.

单词	ExamplesOfCompactSpaces
释义	examples of compact spaces Here are some examples of compact spaces (http://planetmath.org/Compact): • 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 ball of any finite-dimensional normed vector space is compact. This is not true for infinite dimensions; 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 $2^{\\mathbb{N}}$ 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 $x_{k}\ot=y_{k}$ (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 Heine-Borel). This construction can be performed for any finite set, not just {0,1}. • Consider the set $K$ of all functions $f:\\mathbb{R}\\rightarrow[0,1]$ 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. • 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. • 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. • If $H$ is a Hilbert space and $A:H\\rightarrow H$ is a continuous linear operator, then the spectrum of $A$ is a compact subset of $\\mathbb{C}$ . If $H$ is infinite-dimensional, then any compact subset of $\\mathbb{C}$ arises in this manner from some continuous linear operator $A$ on $H$ . • If $\\cal{A}$ is a complex C-algebra which is commutative and contains a one, then the set $X$ of all non-zero algebra homomorphisms $\\phi:\\cal{A}\\rightarrow\\mathbb{C}$ carries a natural topology (the weak- topology) which turns it into a compact Hausdorff space. $\\cal{A}$ is isomorphic 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 product is compact Hausdorff, and a profinite group is a closed subset of such a product. • Any locally compact Hausdorff space 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 $\\mathbb{R}$ is homeomorphic to the circle $S^{1}$ ; the one-point compactification of $\\mathbb{R}^{2}$ is homeomorphic to the sphere $S^{2}$ . 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 bounded totally ordered sets.
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