PlanetMath: wild

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wild

(Definition)

Let be a set in ${\mathbb{R}}^n$ and suppose that is triangulable. ( is triangulable means that when regarded as a space, it has a triangulation.)

If there is a homeomorphism $h: {\mathbb{R}}^n \to {\mathbb{R}}^n$ such that is a polyhedron, we say that is tamely imbedded.

If is triangulable but no such homeomorphism exists is said to be wild.

In ${\mathbb{R}}^2$ every 1-sphere is tamely imbedded. But in ${\mathbb{R}}^3$ there are wild arcs, 1-spheres and 2-spheres.

"wild" is owned by Mathprof.

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Also defines:

tamely imbedded, triangulable

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Cross-references: arcs, polyhedron, homeomorphism, triangulation
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This is version 5 of wild, born on 2007-03-31, modified 2007-04-28.
Object id is 9135, canonical name is Wild.
Accessed 1397 times total.

Classification:

55S37 (Algebraic topology :: Operations and obstructions :: Classification of mappings)

Pending Errata and Addenda

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