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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.)
Zbl 0548.55012
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