PROFINITE TOPOLOGICAL SPACES

Abstract. It is well known [Hoc69, Joy71] that profinite T0-spaces are exactly the spectral spaces. We generalize this result to the category of all topological spaces by showing that the following conditions are equivalent: (1) (X,τ) is a profinite topological space.(2) The T0-reflection of (X,τ) is a profinite T0-space.(3) (X,τ) is a quasi spectral space (in the sense of [BMM08]).(4) (X,τ) admits a stronger Stone topology pi such that (X,τ, pi) is a bitopological quasi spectral space (see Definition 6.1). 1