This book is a course in general topology, intended for students in the
first year of the second cycle (in other words, students in their third university
year). The course was taught during the first semester of the 1979-80
academic year (three hours a week of lecture, four hours a week of guided
work).
Topology is the study of the notions of limit and continuity and thus is, in
principle, very ancient. However, we shall limit ourselves to the origins of the
theory since the nineteenth century. One of the sources of topology is the
effort to clarify the theory of real-valued functions of a real variable: uniform
continuity, uniform convergence, equicontinuity, Bolzano-Weierstrass
theorem (this work is historically inseparable from the attempts to define
with precision what the real numbers are). Cauchy was one of the pioneers in
this direction, but the errors that slip into his work prove how hard it was to
isolate the right concepts. Cantor came along a bit later; his researches into
trigonometric series led him to study in detail sets of points of R (whence the
concepts of open set and closed set in R, which in his work are intermingled
with much subtler concepts).