Generalized Whitney Topologies are Baire

TL;DR

This paper proves that generalized Whitney topologies, including H"older-Whitney and Sobolev-Whitney topologies, possess the Baire property, ensuring the intersection of countably many open dense sets remains dense.

Contribution

It establishes the Baire property for a broad class of Whitney topologies, extending previous results to include H"older-Whitney and Sobolev-Whitney topologies.

Findings

Generalized Whitney topologies are Baire spaces.

Includes H"older-Whitney and Sobolev-Whitney topologies.

Ensures generic properties in these topologies.

Abstract

In this paper we show that certain generalizations of the $C^r$-Whitney topology, which include the H\"older-Whitney and Sobolev-Whitney topologies on smooth manifolds, satisfy the Baire property, to wit, the countable intersection of open and dense sets is dense.