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We continue Mizar formalization of general topology according to the book [16] by Engelking. In the article, we present the final theorem of Section 4.1. Namely, the paper includes the formalization of theorems on the correspondence between the cardinalities of the basis and of some open subcover, and a discreet (closed) subspaces, and the weight of that… (More)

In this article we describe the notion of affinely independent subset of a real linear space. First we prove selected theorems concerning operations on linear combinations. Then we introduce affine independence and prove the equivalence of various definitions of this notion. We also introduce the notion of the affine hull, i.e. a subset generated by a set… (More)

In this paper I present basic properties of block diagonal matrices over a set. In my approach the finite sequence of matrices in a block diagonal matrix is not restricted to square matrices. Moreover, the off-diagonal blocks need not be zero matrices, but also with another arbitrary fixed value. [10] provide the terminology and notation for this paper.

We present the concept and basic properties of the Menger-Urysohn small inductive dimension of topological spaces according to the books [7]. Namely, the paper includes the formalization of main theorems from Sections 1.1 and 1.2. The terminology and notation used here are introduced in the following articles: For simplicity, we adopt the following rules: T… (More)

In this paper we present selected properties of barycentric coordinates in the Euclidean topological space. We prove the topological correspondence between a subset of an affine closed space of E n and the set of vectors created from barycentric coordinates of points of this subset. The terminology and notation used here have been introduced in the… (More)

We introduce linear transformations of Euclidean topological spaces given by a transformation matrix. Next, we prove selected properties and basic arithmetic operations on these linear transformations. Finally, we show that a linear transformation given by an invertible matrix is a homeomorphism. [17] provide the notation and terminology for this paper.

The paper is concerned with a generalization of concepts introduced in [13], i.e. introduced are matrices of linear transformations over a finite-dimensional vector space. Introduced are linear transformations over a finite-dimensional vector space depending on a given matrix of the transformation. Finally, I prove that the rank of linear transformations… (More)

In this paper I present the Kronecker-Capelli theorem which states that a system of linear equations has a solution if and only if the rank of its coefficient matrix is equal to the rank of its augmented matrix. The terminology and notation used in this paper are introduced in the following

In this article we define the notion of abstract simplicial complexes and operations on them. We introduce the following basic notions: simplex, face, vertex, degree, skeleton, subdivision and substructure, and prove some of their properties.