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this paper.

Summary. The aim of the paper is to define some basic notions of restrictions of finite sequences.

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.

Summary. Let S be a subset of the topological Euclidean planeE 2 T. We say that S has Jordan’s property if there exist two non-empty, disjoint and connected subsets G1 and G2 of E2 T such that Sc = G1 ∪ G2 and G1 \ G1 = G2 \ G2 (see [13], [8]). The aim is to prove that the boundaries of some special polygons inE 2 T have this property (see Section 3). Moreover, it is proved that both the interior and the exterior of the boundary of any rectangle inE 2 T is open and connected. MML Identifier:JORDAN1. WWW:http://mizar.org/JFM/Vol4/jordan1.html

Summary. The concept of the minimal distance between two sets in a Euclidean space is introduced and some useful lemmas are proved.

Summary. The aim of the paper is to show that the only subcontinua of the Jordan curve are arcs, the whole curve, and singletons of its points. Additionally, it has been shown that the only subcontinua of the unit interval I are closed intervals. MML Identifier: BORSUK_4.

Summary. We prove a number of auxiliary facts about graphs, mainly about vertex sequences of chains and oriented chains. Then we define a graph to be well-founded if for each vertex in the graph the length of oriented chains ending at the vertex is bounded. A wellfounded graph does not have directed cycles or infinite descending chains. In the second part of the article we prove some auxiliary facts about free algebras and locally-finite algebras. MML Identifier:MSSCYC_1. WWW:http://mizar.org/JFM/Vol8/msscyc_1.html

Summary. On the Euclidean plane Jordan’s curve may be approximated with a polygonal path of sides parallel to coordinate axes, either externally, or internally. The paper deals with the external approximation, and the existence of a Cage – an external polygonal path – is proved.

Summary. This paper is a continuation of [15]. We prove that the family of cosets in the Segre’s product of partial linear spaces remains invariant under automorphisms.