基础拓扑学习题集Elementary Topology Problem Textbook by O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov.pdf
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基础拓扑学习题集Elementary Topology Problem Textbook by O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev, V. M. Kharlamov.pdf
Elementary Topology
Problem Textbook
O. Ya. Viro, O. A. Ivanov,
N. Yu. Netsvetaev, V. M. Kharlamov
Introduction iii
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 deter-
mine 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
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