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Inverse systems and pretopological spaces. (English) Zbl 0566.55008
This paper is the carrying on of ibid. Suppl. 3, 119-126 (1984; Zbl 0548.55012) by the same authors. Given a pretopological space \(S=(X,P)\), we consider the directed set Cov(S) of the interior coverings of S. To any \({\mathcal X}\in Cov(S)\) we associate a pf-space \(S_{{\mathcal X}}\), whose pretopology is given taking for each point \(x\in X\) the principal filter \(\overline{St(x,{\mathcal X})}\). Taking the pf-spaces \(S_{{\mathcal X}}\) as terms, we obtain the inverse system \(\hat S\) of the pretopological space S. For each dimension n, we associate to \(\hat S\) an inverse system of prehomotopy groups \(\Pi_ n(S_{{\mathcal X}},a)\) and an inverse system of singular homology groups \(H_ n(S_{{\mathcal X}})\). The groups \({\check \Pi}{}_ n(S,a)=\lim_{\leftarrow}\Pi_ n(S_{{\mathcal X}},a)\) and Ȟ\({}_ n(S)=\lim_{\leftarrow}H_ n(S_{{\mathcal X}})\) have the characteristical properties of the classical ones.

55Q70 Homotopy groups of special types
55N35 Other homology theories in algebraic topology
54A05 Topological spaces and generalizations (closure spaces, etc.)
Zbl 0548.55012
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