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174
A Borsuk theorem on homotopy types
 Journal of Formalized Mathematics
, 1991
"... Summary. We present a Borsuk’s theorem published first in [1] (compare also [2, pages 119–120]). It is slightly generalized, the assumption of the metrizability is omitted. We introduce concepts needed for the formulation and the proofs of the theorems on upper semicontinuous decompositions, retrac ..."
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Cited by 108 (6 self)
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Summary. We present a Borsuk’s theorem published first in [1] (compare also [2, pages 119–120]). It is slightly generalized, the assumption of the metrizability is omitted. We introduce concepts needed for the formulation and the proofs of the theorems on upper semicontinuous decompositions, retracts, strong deformation retract. However, only those facts that are necessary in the proof have been proved.
Connected spaces
 Journal of Formalized Mathematics
, 1989
"... Summary. This article is a continuation of [3]. We define a neighbourhood of a point and a neighbourhood of a set and prove some facts about them. Then the definitions of a locally connected space and a locally connected set are introduced. Some theorems about locally connected spaces are given (bas ..."
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Cited by 95 (1 self)
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Summary. This article is a continuation of [3]. We define a neighbourhood of a point and a neighbourhood of a set and prove some facts about them. Then the definitions of a locally connected space and a locally connected set are introduced. Some theorems about locally connected spaces are given (based on [2]). We also define a quasicomponent of a point and prove some of its basic properties. MML Identifier: CONNSP_2. WWW:
Bounding boxes for compact sets inE 2
 Journal of Formalized Mathematics
, 1997
"... Summary. We define pseudocompact topological spaces and prove that every compact space is pseudocompact. We also solve an exercise from [14] p.225 that for a topological space X the following are equivalent: • Every continuous real map from X is bounded (i.e. X is pseudocompact). • Every continuous ..."
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Cited by 36 (2 self)
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Summary. We define pseudocompact topological spaces and prove that every compact space is pseudocompact. We also solve an exercise from [14] p.225 that for a topological space X the following are equivalent: • Every continuous real map from X is bounded (i.e. X is pseudocompact). • Every continuous real map from X attains minimum. • Every continuous real map from X attains maximum. Finally, for a compact set in E 2 we define its bounding rectangle and introduce a collection of notions associated with the box.
MooreSmith Convergence
, 2003
"... The paper introduces the concept of a net (a generalized sequence). The goal is to enable the continuation of the translation of [14]. ..."
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Cited by 32 (2 self)
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The paper introduces the concept of a net (a generalized sequence). The goal is to enable the continuation of the translation of [14].
Introduction to the homotopy theory
 Formalized Mathematics
, 1997
"... Summary. The paper introduces some preliminary notions concerning the homotopy theory according to [15]: paths and arcwise connected to topological spaces. The basic operations ..."
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Cited by 20 (3 self)
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Summary. The paper introduces some preliminary notions concerning the homotopy theory according to [15]: paths and arcwise connected to topological spaces. The basic operations
The Jordan’s property for certain subsets of the plane.
 Formalized Mathematics,
, 1992
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The ring of integers, euclidean rings and modulo integers
 Formalized Mathematics
, 1999
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Armstrong’s Axioms
, 2003
"... We present a formalization of the seminal paper by W. W. Armstrong [1] on functional dependencies in relational data bases. The paper is formalized in its entirety including examples and applications. The formalization was done with a routine effort albeit some new notions were defined which simpl ..."
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Cited by 14 (0 self)
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We present a formalization of the seminal paper by W. W. Armstrong [1] on functional dependencies in relational data bases. The paper is formalized in its entirety including examples and applications. The formalization was done with a routine effort albeit some new notions were defined which simplified formulation of some theorems and proofs. The definitive reference to the theory of relational databases is [16], where saturated sets are called closed sets. Armstrong’s “axioms” for functional dependencies are still widely taught at all levels of database design, see for instance [14].