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The Mizar system is one of the pioneering systems aimed at supporting mathematical proof development on a computer that have laid the groundwork for and eventually have evolved into modern interactive proof assistants. We claim that an important milestone in the development of these systems was the creation of organized libraries accumulating all previously available formalized knowledge in such a way that new works could effectively re-use all previously collected notions. In the case of Mizar, the turning point of its development was the decision to start building the Mizar Mathematical Library as a centrally-managed knowledge base maintained together with the formalization language and the verification system. In this paper we show the process of forming this library, the evolution of its design principles, and also present some data showing its current use with the modern version of the Mizar proof checker, but also as a rich corpus of semantically linked mathematical data in various areas including web-based and natural language proof presentation, maths education, and machine learning based automated theorem proving.

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Summary. In the article, I introduce the notions of the compactification of topological spaces and the Alexandroff one point compactification. Some properties of the locally compact spaces and one point compactification are proved. Let X be a topological space and let P be a family of subsets of X. We say that P is compact if and only if: (Def. 1) For every subset U of X such that U ∈ P holds U is compact.Let X be a topological space and let U be a subset of X. We say that U is relatively-compact if and only if: (Def. 2) U is compact.Let X be a topological space. Note that ∅ X is relatively-compact. Let X be a topological space. Observe that there exists a subset of X which is relatively-compact.Let X be a topological space and let U be a relatively-compact subset of X. Observe that U is compact.Let X be a topological space and let U be a subset of X. We introduce U is pre-compact as a synonym of U is relatively-compact.Let X be a non empty topological space. We introduce X is liminallycompact as a synonym of X is locally-compact.Let X be a non empty topological space. Let us observe that X is liminallycompact if and only if: (Def. 3) For every point x of X holds there exists a generalized basis of x which is compact.167

The Mizar language aims to capture mathematical vernacular by providing a rich language for mathematics. From the perspective of a user, the richness of the language is welcome because it makes writing texts more "natural". But for the developer, the richness leads to syntactic complexity, such as dealing with overloading. Recently the Mizar team has been making a fresh approach to the problem of parsing the Mizar language. One aim is to make the language accessible to users and other developers. In this paper we describe these new parsing efforts and some applications thereof, such as large-scale text refactorings, pretty-printing, HTTP parsing services, and normalizations of Mizar texts.

In this article we defined mathematical morphology image processing with set operations. First, we defined Minkowski set operations and proved their properties. Next, we defined basic image processing, dilation and erosion proving basic fact about them [5], [8].