Smooth embeddings of the Long Line and other non-paracompact manifolds into locally convex spaces

TL;DR

The paper demonstrates that all finite-dimensional Hausdorff $C^r$-manifolds, including non-paracompact ones like the Long Line, can be embedded into weakly complete locally convex spaces, and it identifies the minimal size of such spaces.

Contribution

It extends embedding results to non-paracompact manifolds and determines the minimal cardinality of the target space needed for embedding.

Findings

All finite-dimensional Hausdorff $C^r$-manifolds can be embedded into weakly complete locally convex spaces.

The minimal cardinality of the index set for the embedding space is characterized.

Includes non-paracompact manifolds like the Long Line.

Abstract

We show that every finite dimensional Hausdorff (not necessarily paracompact, not necessarily second countable) $C^r$-manifold can be embedded into a weakly complete vector space, i.e. a locally convex topological vector space of the form ${\mathbb R}^I$ for an uncountable index set $I$ and determine the minimal cardinality of $I$ for which such an embedding is possible.