We present a topological characterization for fuzzy regular languages: we show that there is a bijective correspondence between fuzzy regular languages and the set of all clopen fuzzy subsets with finite image in the induced fuzzy topological space of Stone space (Profinite space), and then we give a representation of closed fuzzy subsets in the induced fuzzy topological space via fuzzy regular...

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 ...

For a profinite group, we construct a model structure on profinite spaces with a continuous action. We construct descent spectral sequences for the homotopy groups of the homotopy fixed point space and for the homology of homotopy orbit space which are strongly convergent for an arbitrary profinite group. Our main example is the Galois action on profinite étale topological types of schemes over...

Our main result is that any topological algebra based on a Boolean space is the extended Stone dual space of a certain associated Boolean algebra with additional operations. A particular case of this result is that the profinite completion of any abstract algebra is the extended Stone dual space of the Boolean algebra of recognisable subsets of the abstract algebra endowed with certain residuat...

We construct a model structure on simplicial profinite sets such that the homotopy groups carry a natural profinite structure. This yields a rigid profinite completion functor for spaces and pro-spaces. One motivation is the étale homotopy theory of schemes in which higher profinite étale homotopy groups fit well with the étale fundamental group which is always profinite. We show that the profi...

We define, for any group G, finite approximations ; with this tool, we give a new presentation of the profinite completion b π : G → b G of an abtract group G. We then prove the following theorem : if k is a finite prime field and if V is a k-vector space, then, there is a natural isomorphism between b V (for the underlying additive group structure) and the additive group of the double-dual V ∗...

Abstract This paper is a contribution to understanding what properties should topological algebra on Stone space satisfy be profinite. We reformulate and simplify proofs for some known using syntactic congruences. also clarify the role of various alternative ways describing congruences, namely by finite sets terms compact continuous self mappings algebra.

The sets of closed and closed-normal subgroups of a profinite group carry a natural profinite topology. Through a combination of algebraic and topological methods the size of these subgroup spaces is calculated, and the spaces partially classified up to homeomorphism.