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A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023230/c0232301.png" /> consisting of two points. If both one-point subsets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023230/c0232302.png" /> are open (both are then closed), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023230/c0232303.png" /> is said to be a simple colon. If only one one-point subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023230/c0232304.png" /> is open, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023230/c0232305.png" /> is said to be a connected colon. Finally, if only the empty subset and all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023230/c0232306.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023230/c0232307.png" /> are open, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023230/c0232308.png" /> is called an identified colon; this space — unlike the first two, which are very important though simple — has found no applications.
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A topological space $F$ consisting of two points. If both one-point subsets in $F$ are open (both are then closed), $F$ is said to be a simple colon. If only one one-point subset in $F$ is open, $F$ is said to be a connected colon. Finally, if only the empty subset and all of $F$ in $F$ are open, $F$ is called an identified colon; this space — unlike the first two, which are very important though simple — has found no applications.
  
 
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====References====
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The term two-point discrete space is often applied to the simple colon. Its topological powers are called Cantor cubes. These spaces are universal in two ways: Every zero-dimensional compactum can be imbedded into a Cantor cube of the same weight, and every compactum can be obtained as the continuous image of a closed set of a Cantor cube of the same weight. These facts generalize well-known results on the Cantor cube of countable weight, the Cantor set. The connected colon is also known as Sierpiński's space. Its topological powers are called Alexandrov cubes; they are universal in that they contain all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023230/c0232309.png" />-spaces topologically.
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The term two-point discrete space is often applied to the simple colon. Its topological powers are called Cantor cubes. These spaces are universal in two ways: Every zero-dimensional compactum can be imbedded into a Cantor cube of the same weight, and every compactum can be obtained as the continuous image of a closed set of a Cantor cube of the same weight. These facts generalize well-known results on the Cantor cube of countable weight, the Cantor set. The connected colon is also known as Sierpiński's space. Its topological powers are called Alexandrov cubes; they are universal in that they contain all $T_0$-spaces topologically.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , PWN  (1977)  (Translated from Polish)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , PWN  (1977)  (Translated from Polish)</TD></TR></table>

Revision as of 18:43, 16 April 2014

A topological space $F$ consisting of two points. If both one-point subsets in $F$ are open (both are then closed), $F$ is said to be a simple colon. If only one one-point subset in $F$ is open, $F$ is said to be a connected colon. Finally, if only the empty subset and all of $F$ in $F$ are open, $F$ is called an identified colon; this space — unlike the first two, which are very important though simple — has found no applications.

References

[1] P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)


Comments

The term two-point discrete space is often applied to the simple colon. Its topological powers are called Cantor cubes. These spaces are universal in two ways: Every zero-dimensional compactum can be imbedded into a Cantor cube of the same weight, and every compactum can be obtained as the continuous image of a closed set of a Cantor cube of the same weight. These facts generalize well-known results on the Cantor cube of countable weight, the Cantor set. The connected colon is also known as Sierpiński's space. Its topological powers are called Alexandrov cubes; they are universal in that they contain all $T_0$-spaces topologically.

References

[a1] R. Engelking, "General topology" , PWN (1977) (Translated from Polish)
How to Cite This Entry:
Colon. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Colon&oldid=31795
This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article