Theorem

Whitney Immersion Theorem

Topology

The Whitney Immersion Theorem, named after Hassler Whitney, states that for a dimension m greater than one, any smooth m-dimensional manifold that is Hausdorff and second-countable has a one-to-one immersion into Euclidean 2m-dimensional space, and can be immersed, not necessarily one-to-one, into (2m-1)-dimensional space; every such manifold can likewise be immersed in the (2m-1)-dimensional sphere, which removes the restriction to m greater than one. The weaker version of the result, into (2m+1)-dimensional space, follows simply from a dimension-counting transversality argument, since two m-dimensional manifolds sitting inside 2m-dimensional Euclidean space intersect generically only in a zero-dimensional set.

Facts

Statement

For m > 1, any smooth m-dimensional manifold (Hausdorff and second-countable) has a one-to-one immersion in Euclidean 2m-space. 1

Classification

Statement Form

Inequality 1

Connections

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Manifold

Has Statement Form

Inequality

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

In Branch

Differential Topology

Proved By

Hassler Whitney

Source Whitney immersion theorem (Wikipedia)

Sources

1. Whitney immersion theorem (Wikipedia)

Lead paragraph, sentence 1
any smooth m-dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean 2m-space
Proved By: Hassler Whitney, Lead paragraph
In differential topology, the Whitney immersion theorem (named after Hassler Whitney) states that for

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Whitney Immersion Theorem

Associated With

Has Statement Form

In Branch

Proved By