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Elementary Topology Problem Textbook O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov
Introduction
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Dedicated to the memory of Vladimir Abramovich Rokhlin (1919–1984) – our teacher
Introduction The subject of the book, Elementary Topology Elementary means close to elements, basics. It is impossible to determine precisely, once and for all, which topology is elementary, and which is not. The elementary part of a subject is the part with which an expert starts to teach a novice. We suppose that our student is ready to study topology. So, we do not try to win her or his attention and benevolence by hasty and obscure stories about misterious and attractive things such as the Klein bottle.1 All in good time: the Klein bottle will appear in its turn. However, we start with what a topological space is. That is, we start with general topology. General topology became a part of the general mathematical language long ago. It teaches one to speak clearly and precisely about things related to the idea of continuity. It is needed not only in order to explain what, finally, the Klein bottle is. This is also a way to introduce geometrical images into any area of mathematics, no matter how far from geometry the area may be at first glance. As an active research area, general topology is practically completed. A permanent usage in the capacity of a general mathematical language has polished its system of definitions and theorems. Nowadays, study of general topology indeed resembles rather a study of a language than a study of mathematics: one has to learn many new words, while the proofs of the majority of theorems are extremely simple. But the quantity of the theorems is huge. This comes as no surprise because they play the role of rules that regulate usage of words. The book consists of two parts. General topology is the subject of the first one. The second par