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A [[Metric space|metric space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093430/t0934301.png" /> that, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093430/t0934302.png" />, can be represented as the union of a finite number of sets with diameters smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093430/t0934303.png" />. An equivalent condition is the following: For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093430/t0934304.png" /> there exists in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093430/t0934305.png" /> a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093430/t0934307.png" />-net, i.e. a finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093430/t0934308.png" /> such that the distance of each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093430/t0934309.png" /> from some point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093430/t09343010.png" /> is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093430/t09343011.png" />. Totally-bounded spaces are those, and only those, metric spaces that can be represented as subspaces of compact metric spaces (cf. [[Compact space|Compact space]]). The metric totally-bounded spaces, considered as topological spaces, exhaust all regular spaces (cf. [[Regular space|Regular space]]) with a countable base. A subspace of a Euclidean space is totally bounded if and only if it is bounded. The converse is not true: An infinite space in which the distance between any two points is one, as well as a sphere and a ball of a Hilbert space, are bounded, but not totally bounded, metric spaces. The significance of the concept of a totally-bounded space may be illustrated by the following theorem: A metric space is a [[Compactum|compactum]] if and only if it is totally bounded and complete. The metric completion of a metric totally-bounded space is compact. The image of a totally-bounded space under a uniformly continuous mapping is a totally-bounded space.
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A [[Metric space|metric space]] $X$ that, for any $\epsilon>0$, can be represented as the union of a finite number of sets with diameters smaller than $\epsilon$. An equivalent condition is the following: For each $\epsilon>0$ there exists in $X$ a finite $\epsilon$-net, i.e. a finite set $A$ such that the distance of each point of $X$ from some point of $A$ is less than $\epsilon$. Totally-bounded spaces are those, and only those, metric spaces that can be represented as subspaces of compact metric spaces (cf. [[Compact space|Compact space]]). The metric totally-bounded spaces, considered as topological spaces, exhaust all regular spaces (cf. [[Regular space|Regular space]]) with a countable base. A subspace of a Euclidean space is totally bounded if and only if it is bounded. The converse is not true: An infinite space in which the distance between any two points is one, as well as a sphere and a ball of a Hilbert space, are bounded, but not totally bounded, metric spaces. The significance of the concept of a totally-bounded space may be illustrated by the following theorem: A metric space is a [[Compactum|compactum]] if and only if it is totally bounded and complete. The metric completion of a metric totally-bounded space is compact. The image of a totally-bounded space under a uniformly continuous mapping is a totally-bounded space.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Hausdorff,  "Grundzüge der Mengenlehre" , Leipzig  (1914)  (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.S. Aleksandrov,  "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Hausdorff,  "Grundzüge der Mengenlehre" , Leipzig  (1914)  (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.S. Aleksandrov,  "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR></table> <table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR></table>
 
 
 
 
 
 
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====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR></table>
 

Latest revision as of 13:16, 12 December 2013

A metric space $X$ that, for any $\epsilon>0$, can be represented as the union of a finite number of sets with diameters smaller than $\epsilon$. An equivalent condition is the following: For each $\epsilon>0$ there exists in $X$ a finite $\epsilon$-net, i.e. a finite set $A$ such that the distance of each point of $X$ from some point of $A$ is less than $\epsilon$. Totally-bounded spaces are those, and only those, metric spaces that can be represented as subspaces of compact metric spaces (cf. Compact space). The metric totally-bounded spaces, considered as topological spaces, exhaust all regular spaces (cf. Regular space) with a countable base. A subspace of a Euclidean space is totally bounded if and only if it is bounded. The converse is not true: An infinite space in which the distance between any two points is one, as well as a sphere and a ball of a Hilbert space, are bounded, but not totally bounded, metric spaces. The significance of the concept of a totally-bounded space may be illustrated by the following theorem: A metric space is a compactum if and only if it is totally bounded and complete. The metric completion of a metric totally-bounded space is compact. The image of a totally-bounded space under a uniformly continuous mapping is a totally-bounded space.

References

[1] J.L. Kelley, "General topology" , Springer (1975)
[2] F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))
[3] P.S. Aleksandrov, "Einführung in die Mengenlehre und die Theorie der reellen Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[4] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[a1] R. Engelking, "General topology" , Heldermann (1989)
How to Cite This Entry:
Totally-bounded space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Totally-bounded_space&oldid=15853
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article