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**1 - 6**of**6**### Tietze Extension Theorem for n-dimensional Spaces

, 2014

"... Summary. In this article we prove the Tietze extension theorem for an arbitrary convex compact subset of E n with a non-empty interior. This theorem states that, if T is a normal topological space, X is a closed subset of T , and A is a convex compact subset of E n with a non-empty interior, then a ..."

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Summary. In this article we prove the Tietze extension theorem for an arbitrary convex compact subset of E n with a non-empty interior. This theorem states that, if T is a normal topological space, X is a closed subset of T , and A is a convex compact subset of E n with a non-empty interior, then a continuous function f : X → A can be extended to a continuous function g : T → E n . Additionally we show that a subset A is replaceable by an arbitrary subset of a topological space that is homeomorphic with a convex compact subset of E n with a non-empty interior. This article is based on MSC: 54A05 03B35

### Towards a Coherent Repository of Knowledge

, 2007

"... In this paper, we describe the development of the Mizar Mathematical Library used to formally prove the Jordan Curve Theorem. More general issues of knowledge reusability are also raised with special attention paid to the logical equivalence between some formal apparatus of topology and real analy ..."

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In this paper, we describe the development of the Mizar Mathematical Library used to formally prove the Jordan Curve Theorem. More general issues of knowledge reusability are also raised with special attention paid to the logical equivalence between some formal apparatus of topology and real analysis.

### A Proof of the Jordan Curve Theorem via the Brouwer Fixed Point Theorem

"... Abstract – The aim of the paper is to report on MIZAR codification of the Jordan curve theorem, a theorem chosen as a challenge to be completely verified using formal methods at the time when they started being commonly used. Formalization was done based on proofs taken from the literature, where th ..."

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Abstract – The aim of the paper is to report on MIZAR codification of the Jordan curve theorem, a theorem chosen as a challenge to be completely verified using formal methods at the time when they started being commonly used. Formalization was done based on proofs taken from the literature, where theorems mentioned in the title of the paper from ”Brouwer’s Fixed Point Theorem and the

### STUDIES IN LOGIC, GRAMMAR AND RHETORIC 10 (23) 2007 Towards a Mizar Mathematical Library in OMDoc Format

"... Abstract. Mizar is one of largest libraries of formalized mathematics. The language of the library is highly optimized for authoring by humans. Like in natural languages, the meaning of an expression is influenced by its (mathematical) context in a way that is natural to humans, but hard to specify ..."

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Abstract. Mizar is one of largest libraries of formalized mathematics. The language of the library is highly optimized for authoring by humans. Like in natural languages, the meaning of an expression is influenced by its (mathematical) context in a way that is natural to humans, but hard to specify for machine manipulation. From this point of view, it may be considered as locked up in an arcane file format. Indeed, the Mizar system itself is currently the only system that can reliably operate on the Mizar library. This paper presents an experiment of using the Mizar system to transform the Mizar library into the OMDoc format (Open Mathematical Documents), an XML-based representation format for mathematical knowledge that is geared towards making formula structure and context dependencies explicit. We expect the result of this experiment: an OMDoc version of the Mizar library to enhance system support for formal mathematical libraries. 1