Difference between revisions of "Dispersive space"
From Encyclopedia of Mathematics
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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 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. | + | 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=30591
Dispersive space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dispersive_space&oldid=30591
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