This book arose from lecture notes of a course given to first and s
year students at the Independent University of Moscow.
Topology is a very beautiful science. It is the bridge betwe
ometry and algebra. Its ideas and images play a key role in alm
of modern mathematics: in differential equations, mechanics, co
analysis, algebraic geometry, functional analysis, mathematica
quantum physics, representation theory, and even—in a surpri
modified form—in number theory, combinatorics, and complexit
In recent years most of the new ideas in mathematics ar
topology from geometrical images and were then formalized an
ried over to more algebraic fields. For this reason a sound know
of topology is necessary to any research mathematician. Un
nately, in Russia and many other countries, topology is not inc
even today, in the basic curriculum of mathematical departme
most universities. Serious teachers of the other disciplines mu
clude various fragments of topology in their courses, but the s
who studies Stokes' formula in the calculus, the argument pri
and Riemann surfaces in complex analysis, the principle of co
ing maps and the index of singular points of vector fields in di
tial equations, the Euler characteristic in combinatorics, stable r
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