Two extensions of the Stone Duality to the category of zero-dimensional Hausdorff spaces

Abstract

Extending the Stone Duality Theorem, we describe two categories which are dually equivalent to the category ZHaus of zero-dimensional Hausdorff spaces and continuous maps. We find as well two categories which are dually equivalent to the category ZComp of all zero-dimensional Hausdorff compactifications of zero-dimensional Hausdorff spaces.

Extensions of the Stone Duality to the category BooleSp

“…In Section 3 we introduce the notion of Boolean ldz-algebra and present our first new duality theorem for the category BooleSp (see Theorem 3.3). It extends the Stone Duality Theorem and is obtained with the help of our duality theorem [5,Theorem 3.15]. After that, using it, we give a new proof of the Dimov Duality Theorem for the category BooleSp (see 3.5 and Theorem 3.4).…”

“…Later on, G. Dimov [3,4] extended the Stone Duality to the category BooleSp of Boolean spaces and continuous maps. Finally, in [5], we extended the Stone Duality to the category ZDHaus of zero-dimensional Hausdorff spaces and continuous maps.…”

“…In this paper, which can be regarded as a continuation of [5], we show how the Dimov Duality Theorem mentioned above can be derived from one of our general duality theorems for the category ZDHaus proved in [5]. Moreover, with the help of our results from [5], we obtain two new extensions of the Stone Duality Theorem to the category BooleSp, one of which is in the spirit of the recent Duality Theorem of Bezhnanishvili, Morandi and Olberding [2] for the category Tych of Tychonoff spaces and continuous maps.…”

“…In Section 2 we collect all preliminary facts which are needed for the exposition of our results. In particular, we briefly recall and fix the notation pertaining to all, the Stone Duality, the Tarski Duality and our two duality theorems from [5] with which we extended the Stone Duality Theorem to the category ZDHaus. In Section 3 we introduce the notion of Boolean ldz-algebra and present our first new duality theorem for the category BooleSp (see Theorem 3.3).…”

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In [5], extending the Stone Duality Theorem, we proved two duality theorems for the category ZDHaus of zero-dimensional Hausdorff spaces and continuous maps. Now we derive from them the extension of the Stone Duality Theorem to the category BooleSp of zero-dimensional locally compact Hausdorff spaces and continuous maps obtained in [4], as well as two new duality theorems for the category BooleSp.

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