On the Lusternik–Schnirelmann category of Peano continua

Abstract

We define the LS-category catg by means of covers of a space by general subsets, and show that this definition coincides with the classical Lusternik–Schnirelmann category for compact metric ANR spaces. We apply this result to give short dimension theoretic proofs of the Grossman–Whitehead theorem and Dranishnikovʼs theorem. We compute catg for some fractal Peano continua such as Menger spaces and Pontryagin surfaces.