Peano space: Difference between revisions
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* There is a continuous surjective map from the closed unit interval <math>[0,1]</math> to the given space, and the space is [[Hausdorff space|Hausdorff]] | * There is a continuous surjective map from the closed unit interval <math>[0,1]</math> to the given space, and the space is [[Hausdorff space|Hausdorff]] | ||
* The space is [[Hausdorff space|Hausdorff]], [[compact space|compact]], [[connected space|connected]], [[metrizable space|metrizable]], and [[ | * The space is [[Hausdorff space|Hausdorff]], [[compact space|compact]], [[connected space|connected]], [[metrizable space|metrizable]], and [[locally connected space|locally connected]] | ||
===Equivalence of definitions=== | ===Equivalence of definitions=== | ||
The two definitions are equivalent via the [[Hahn-Mazurkiewicz theorem]]. | The two definitions are equivalent via the [[Hahn-Mazurkiewicz theorem]]. | ||
Revision as of 22:02, 1 December 2007
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
Definition
Symbol-free definition
A Peano space is a space which satisfies the following equivalent conditions:
- There is a continuous surjective map from the closed unit interval to the given space, and the space is Hausdorff
- The space is Hausdorff, compact, connected, metrizable, and locally connected
Equivalence of definitions
The two definitions are equivalent via the Hahn-Mazurkiewicz theorem.