On the Lusternik-Schnirelmann category of Peano continua

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.