Difference between revisions of "Irreducible continuum"
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| + | $#C+1 = 4 : ~/encyclopedia/old_files/data/I052/I.0502580 Irreducible continuum | ||
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| + | A non-degenerate [[Continuum|continuum]] that is irreducible between a certain pair of points, that is, does not contain any proper subcontinuum containing these points. | ||
====Comments==== | ====Comments==== | ||
| − | An example is the famous curve of | + | An example is the famous curve of $ \sin ( 1 / x ) $: |
| + | it is the subset $ \{ 0 \} \times [ - 1 , 1 ] \cup \{ {( x, \sin ( 1 / x ) ) } : {0 < x \leq 1 } \} $ | ||
| + | of the plane. This curve is irreducible between the points $ ( 0 , 0 ) $ | ||
| + | and $ ( 1 , \sin 1 ) $. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Kuratowski, "Topology" , '''2''' , Acad. Press (1968) (Translated from French)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Kuratowski, "Topology" , '''2''' , Acad. Press (1968) (Translated from French)</TD></TR></table> | ||
Latest revision as of 22:13, 5 June 2020
A non-degenerate continuum that is irreducible between a certain pair of points, that is, does not contain any proper subcontinuum containing these points.
Comments
An example is the famous curve of $ \sin ( 1 / x ) $: it is the subset $ \{ 0 \} \times [ - 1 , 1 ] \cup \{ {( x, \sin ( 1 / x ) ) } : {0 < x \leq 1 } \} $ of the plane. This curve is irreducible between the points $ ( 0 , 0 ) $ and $ ( 1 , \sin 1 ) $.
References
| [a1] | K. Kuratowski, "Topology" , 2 , Acad. Press (1968) (Translated from French) |
How to Cite This Entry:
Irreducible continuum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irreducible_continuum&oldid=47433
Irreducible continuum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irreducible_continuum&oldid=47433
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