Difference between revisions of "Component of a space"
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| − | A connected subset '''''C''''' of a topological space '''''X''''' with the following property: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024240/c0242403.png" /> is a connected subset such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024240/c0242404.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024240/c0242405.png" />. The components of a space are disjoint. Every non-empty connected subset is contained in exactly one component. If | + | A connected subset '''''C''''' of a topological space '''''X''''' with the following property: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024240/c0242403.png" /> is a connected subset such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024240/c0242404.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024240/c0242405.png" />. The components of a space are disjoint. Every non-empty connected subset is contained in exactly one component. If '''''C''''' is a component of a space '''''X''''' and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024240/c0242408.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024240/c0242409.png" /> is a component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024240/c02424010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024240/c02424011.png" /> is a monotone continuous mapping onto, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024240/c02424012.png" /> is a component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024240/c02424013.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024240/c02424014.png" /> is a component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024240/c02424015.png" />. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Kuratowski, "Topology" , '''2''' , Acad. Press (1968) (Translated from French)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Kuratowski, "Topology" , '''2''' , Acad. Press (1968) (Translated from French)</TD></TR></table> | ||
Revision as of 20:31, 25 March 2011
A connected subset C of a topological space X with the following property: If 
is a connected subset such that 
, then 
. The components of a space are disjoint. Every non-empty connected subset is contained in exactly one component. If C is a component of a space X and 
, then 
is a component of 
. If 
is a monotone continuous mapping onto, then 
is a component of 
if and only if 
is a component of 
.
References
| [1] | K. Kuratowski, "Topology" , 2 , Acad. Press (1968) (Translated from French) |
How to Cite This Entry:
Component of a space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Component_of_a_space&oldid=19326
Component of a space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Component_of_a_space&oldid=19326
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article