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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.

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