Difference between revisions of "Knaster continuum"
From Encyclopedia of Mathematics
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''hereditarily indecomposable continuum'' | ''hereditarily indecomposable continuum'' | ||
| − | A continuum each subcontinuum of which is indecomposable. A space | + | A [[continuum]] each subcontinuum of which is [[Indecomposable continuum|indecomposable]]. A space $X$ is called indecomposable if it is connected and cannot be represented as the union of two closed connected proper subsets of it. |
| − | The first proof of the existence of such a continuum was given by B. Knaster [[#References|[1]]]. In the space of all subcontinua of the ordinary square | + | The first proof of the existence of such a continuum was given by B. Knaster [[#References|[1]]]. In the space of all subcontinua of the ordinary square $I^2$, the set of all Knaster continua is an [[Everywhere-dense set|everywhere-dense]] [[G-delta|$G_\delta$]]-set [[#References|[2]]]. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> B. Knaster, "Un continu dont tout sous-continu est indécomposable" ''Fund. Math.'' , '''3''' (1922) pp. 247–286</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Mazurkiewicz, "Sur les continus absolument indécomposables" ''Fund. Math.'' , '''16''' (1930) pp. 151–159</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> B. Knaster, "Un continu dont tout sous-continu est indécomposable" ''Fund. Math.'' , '''3''' (1922) pp. 247–286 {{ZBL|48.0212.01}}</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> S. Mazurkiewicz, "Sur les continus absolument indécomposables" ''Fund. Math.'' , '''16''' (1930) pp. 151–159 {{ZBL|56.1135.02}}</TD></TR> | ||
| + | </table> | ||
====Comments==== | ====Comments==== | ||
| − | See also [[ | + | See also [[Pseudo-arc]] and [[Hereditarily indecomposable continuum]]. |
Latest revision as of 16:42, 10 January 2015
2020 Mathematics Subject Classification: Primary: 54F15 [MSN][ZBL]
hereditarily indecomposable continuum
A continuum each subcontinuum of which is indecomposable. A space $X$ is called indecomposable if it is connected and cannot be represented as the union of two closed connected proper subsets of it.
The first proof of the existence of such a continuum was given by B. Knaster [1]. In the space of all subcontinua of the ordinary square $I^2$, the set of all Knaster continua is an everywhere-dense $G_\delta$-set [2].
References
| [1] | B. Knaster, "Un continu dont tout sous-continu est indécomposable" Fund. Math. , 3 (1922) pp. 247–286 Zbl 48.0212.01 |
| [2] | S. Mazurkiewicz, "Sur les continus absolument indécomposables" Fund. Math. , 16 (1930) pp. 151–159 Zbl 56.1135.02 |
Comments
See also Pseudo-arc and Hereditarily indecomposable continuum.
How to Cite This Entry:
Knaster continuum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Knaster_continuum&oldid=36191
Knaster continuum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Knaster_continuum&oldid=36191
This article was adapted from an original article by L.G. Zambakhidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article