The book is based on a topology course taught to first- and second-year students at NMU, as well as to American students as part of the Math in Moscow program. The first part is a general introduction to topology, with an emphasis on low-dimensional geometric objects (graphs, surfaces, curves on a plane, nodes) and their invariants (Eulerian characteristic, degree of mapping of a circle, degree of a point relative to a curve, fundamental group). The second part is an introduction to algebraic topology, including homotopy groups, cellular, simplicial and singular homology, along with such classics as Poincaré duality, obstacle theory, Gurevich, Hopf-Whitney, Lefschetz theorems, Eilenberg-MacLane spaces, vector bundles. students and university teachers.