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Difference between revisions of "Dispersive space"

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====Comments====
 
====Comments====
Some authors call spaces as above totally disconnected. However, a space is commonly called totally disconnected if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033410/d0334101.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033410/d0334102.png" /> there is a closed and open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033410/d0334103.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033410/d0334104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033410/d0334105.png" />. A closed and open set is also called a clopen set.
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Some authors call spaces as above totally disconnected. However, a space is commonly called totally disconnected if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033410/d0334101.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033410/d0334102.png" /> there is a [[Open-closed set|closed and open set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033410/d0334103.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033410/d0334104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033410/d0334105.png" />. A closed and open set is also called a clopen set.
  
 
See also [[Totally-disconnected space|Totally-disconnected space]].
 
See also [[Totally-disconnected space|Totally-disconnected space]].

Revision as of 18:36, 30 September 2013

hereditarily disconnected space

A topological space which contains no connected sets with more than one point.


Comments

Some authors call spaces as above totally disconnected. However, a space is commonly called totally disconnected if for all in there is a closed and open set such that and . A closed and open set is also called a clopen set.

See also Totally-disconnected space.

References

[a1] K. Kuratowski, "Topology" , 1–2 , Acad. Press (1966–1968) (Translated from French)
[a2] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
How to Cite This Entry:
Dispersive space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dispersive_space&oldid=16150
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