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Definitions and Properties of the Join and Meet of Subsets
, 1996
"... This paper is the continuation of formalization of [4]. The definitions of meet and join of subsets of relational structures are introduced. The properties of these notions are proved. MML Identifier: YELLOW_4 ..."
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Cited by 12 (2 self)
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This paper is the continuation of formalization of [4]. The definitions of meet and join of subsets of relational structures are introduced. The properties of these notions are proved. MML Identifier: YELLOW_4
On the characterizations of compactness
 Journal of Formalized Mathematics
"... Summary. In the paper we show equivalence of the convergence of filters on a topological space and the convergence of nets in the space. We also give, five characterizations of compactness. Namely, for any topological space T we proved that following condition are equivalent: • T is compact, • every ..."
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Cited by 8 (1 self)
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Summary. In the paper we show equivalence of the convergence of filters on a topological space and the convergence of nets in the space. We also give, five characterizations of compactness. Namely, for any topological space T we proved that following condition are equivalent: • T is compact, • every ultrafilter on T is convergent, • every proper filter on T has cluster point, • every net in T has cluster point, • every net in T has convergent subnet, • every Cauchy net in T is convergent.
On ordering of bags
 Journal of Formalized Mathematics
"... Summary. We present a Mizar formalization of chapter 4.4 of [8] devoted to special orderings in additive monoids to be used for ordering terms in multivariate polynomials. We have extended the treatment to the case of infinite number of variables. It turns out that in such case admissible orderings ..."
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Cited by 6 (1 self)
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Summary. We present a Mizar formalization of chapter 4.4 of [8] devoted to special orderings in additive monoids to be used for ordering terms in multivariate polynomials. We have extended the treatment to the case of infinite number of variables. It turns out that in such case admissible orderings are not necessarily well orderings. MML Identifier:BAGORDER. WWW:http://mizar.org/JFM/Vol14/bagorder.html
A Compendium of Continuous Lattices in MIZAR  Formalizing recent mathematics
, 2002
"... This paper reports on the Mizar formalization of the theory of continuous lattices as presented in A Compendium of Continuous Lattices, [25]. By the Mizar formalization we mean a formulation of theorems, de nitions, and proofs written in the Mizar language whose correctness is veri ed by the Mizar ..."
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Cited by 5 (0 self)
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This paper reports on the Mizar formalization of the theory of continuous lattices as presented in A Compendium of Continuous Lattices, [25]. By the Mizar formalization we mean a formulation of theorems, de nitions, and proofs written in the Mizar language whose correctness is veri ed by the Mizar processor. This eort was originally motivated by the question of whether or not the Mizar system was suciently developed for the task of expressing advanced mathematics. The current state of the formalization57 Mizar articles written by 16 authors indicates that in principle the Mizar system has successfully met the challenge. To our knowledge it is the most sizable eort aimed at mechanically checking some substantial and relatively recent eld of advanced mathematics. However, it does not mean that doing mathematics in Mizar is as simple as doing mathematics traditionally (if doing mathematics is simple at all). The work of formalizing the material of [25] has: (i) prompted many improvements of the Mizar proof checking system; (ii) caused numerous revisions of the the Mizar data base; and (iii) contributed to the \to do" list of further changes to the Mizar system.
Continuous lattices of maps between T0 spaces
 Journal of Formalized Mathematics
, 1999
"... Let I be a set and let J be a relational structure yielding many sorted set indexed by I. We introduce Iprod POS J as a synonym of ∏J. Let I be a set and let J be a relational structure yielding nonempty many sorted set indexed by I. Observe that Iprod POS J is constituted functions. Let I be a se ..."
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Let I be a set and let J be a relational structure yielding many sorted set indexed by I. We introduce Iprod POS J as a synonym of ∏J. Let I be a set and let J be a relational structure yielding nonempty many sorted set indexed by I. Observe that Iprod POS J is constituted functions. Let I be a set and let J be a topological space yielding nonempty many sorted set indexed by I. We introduce Iprod TOP J as a synonym of ∏J. Let X, Y be non empty topological spaces. The functor [X → Y] yields a non empty strict relational structure and is defined as follows: (Def. 1) [X → Y] = [X → ΩY]. Let X, Y be non empty topological spaces. One can check that [X → Y] is reflexive, transitive, and constituted functions. Let X be a non empty topological space and let Y be a non empty T0 topological space. Observe that [X → Y] is antisymmetric. One can prove the following three propositions: (1) Let X, Y be non empty topological spaces and a be a set. Then a is an element of [X → Y]