Topology is a subject that has only recently been introduced into the curriculum of mathematics departments. However, it does, in our opinion, play quite a considerable role with respect to university mathematical analysis, differential equations, differential geometry, mechanics, and functional analysis that correspond to the modern state of these disciplines without involving topologicl concepts. It is therefore essential to acquaint students with topological methods of research as soon as they start their first university courses. Having lectured on topology for some time to first year university students, we realised that there is a great need for a textbook that is comprehensible to students who have a minimal knowledge of mathematics (i.e., they are cognizant of general set theory, general algebra, the elements of linear algebra and mathematical analysis) and that will introduce a reader to the basic ideas underlying modern topology. At the same time, the book should contain a certain volume of topological concepts and methods. This toxtboox is one of the many possible variants of a first course in topology and is written in accordance with both the uthors` preferences and their experience as lecturers and researchers. It deals with those areas of topology that are most closely related to fundamental courses in general mathematics and applications. The material leaves a lecturer a free choice as to how he or she may want to design his or her own topology course and seminar casses.