Topological linear compactness for Grothendieck categories. Theorem of Tychonoff. Applications to coalgebras

We show the Tychonoff's theorem for a Grothendieck category with a set of small projective generators. Strictly quasi-finite objects for semiartinian Grothendieck categories are characterized. We apply these results to the study of the Morita duality of dual algebra of a coalgebra

Proofs of Tychonoff's Theorem

This bachelor thesis is devoted to four different proofs of Tychonoff's Theorem. The first proof is based on definitions of compact topological space and product topology. The second proof is a construction of convergent subnet of an arbitrary net in a product of compact spaces. The third proof uses the fact that topological space is compact if and only if every universal net is convergent. The last proof is based on characterization of compact spaces using systems of closed subsets with the finite intersection property. 1Tato bakalářská práce zpracovává čtyři různé důkazy Tichonovovy věty. První důkaz vychází z definic kompaktního topologického prostoru a součinové topologie. Druhý důkaz je konstrukcí kon- vergentního podnetu libovolného netu v součinu kompaktních prostorů. Třetí důkaz využívá toho, že topologický prostor je kompaktní právě tehdy, když každý univerzální net je konvergentní. Poslední důkaz vychází ze souvislosti centrovaného systému uzavřených množin a kompaktnosti. 1Department of Mathematical AnalysisKatedra matematické analýzyMatematicko-fyzikální fakultaFaculty of Mathematics and Physic

An abstract disintegration theorem

A Strassen-type disintegration theorem for convex cones with localized order structure is proved. As an example a flow theorem for infinite networks is given

Products of effective topological spaces and a uniformly computable Tychonoff Theorem

This article is a fundamental study in computable analysis. In the framework of Type-2 effectivity, TTE, we investigate computability aspects on finite and infinite products of effective topological spaces. For obtaining uniform results we introduce natural multi-representations of the class of all effective topological spaces, of their points, of their subsets and of their compact subsets. We show that the binary, finite and countable product operations on effective topological spaces are computable. For spaces with non-empty base sets the factors can be retrieved from the products. We study computability of the product operations on points, on arbitrary subsets and on compact subsets. For the case of compact sets the results are uniformly computable versions of Tychonoff's Theorem (stating that every Cartesian product of compact spaces is compact) for both, the cover multi-representation and the "minimal cover" multi-representation