DSpace Zespół:

On the Boundary and Derivative of a Set

2026-11-04T13:46:22Z

Tytuł: On the Boundary and Derivative of a Set Autorzy: Grabowski, Adam Abstrakt: This is the first Mizar article in a series aiming at a complete formalization of the textbook “General Topology” by Engelking [7]. We cover the first part of Section 1.3, by defining such notions as a derivative of a subset A of a topological space (usually denoted by Ad, but Der A in our notation), the derivative and the boundary of families of subsets, points of accumulation and isolated points. We also introduce dense-in-itself, perfect and scattered topological spaces and formulate the notion of the density of a space. Some basic properties are given as well as selected exercises from [7].

Spaces of Pencils, Grassmann Spaces, and Generalized Veronese Spaces

2026-11-04T13:14:22Z

Tytuł: Spaces of Pencils, Grassmann Spaces, and Generalized Veronese Spaces Autorzy: Naumowicz, Adam Abstrakt: In this paper we construct several examples of partial linear spaces. First, we define two algebraic structures, namely the spaces of k-pencils and Grassmann spaces for vector spaces over an arbitrary field. Then we introduce the notion of generalized Veronese spaces following the definition presented in the paper [8] by Naumowicz and Prażmowski. For all spaces defined, we state the conditions under which they are not degenerated to a single line.

On the Characterization of Collineations of the Segre Product of Strongly Connected Partial Linear Spaces

2026-11-04T12:42:01Z

Tytuł: On the Characterization of Collineations of the Segre Product of Strongly Connected Partial Linear Spaces Autorzy: Naumowicz, Adam Abstrakt: In this paper we characterize the automorphisms (collineations) of the Segre product of partial linear spaces. In particular, we show that if all components of the product are strongly connected, then every collineation is determined by a set of isomorphisms between its components. The formalization follows the ideas presented in the Journal of Geometry paper [16] by Naumowicz and Prażmowski.

The Quaternion Numbers

2026-11-04T12:21:48Z

Tytuł: The Quaternion Numbers Autorzy: Liang, Xiquan; Ge, Fuguo Abstrakt: In this article, we define the set H of quaternion numbers as the set of all ordered sequences q = where x,y,w and z are real numbers. The addition, difference and multiplication of the quaternion numbers are also defined. We define the real and imaginary parts of q and denote this by x = ℜ(q), y = ℑ1(q), w = ℑ2(q), z = ℑ3(q). We define the addition, difference, multiplication again and denote this operation by real and three imaginary parts. We define the conjugate of q denoted by q*' and the absolute value of q denoted by |q|. We also give some properties of quaternion numbers.