Idézet:Math is a magical subject. We obtain different results when we define different spaces, and their scopes vary accordingly. The metric space solves the problem of convergence and similarity and can prove the existence and uniqueness of solutions using the fixed point theorem. The normed linear space addresses the problem of closure under elementary linear operations and can solve application problems using four main theorems: the Hahn-Banach theorem, the uniform boundedness theorem, the open mapping theorem, and the closed graph theorem. Sobolev space can solve weak solutions of differential equations. However, with the continuous development of mathematical research, it became difficult for people to limit themselves to metric space alone. As researchers began to expand the scope of their investigation, the discipline of topology emerged. Topology has its own function in our everyday study and life.

Topology concerns the study of spatial objects such as curves, surfaces, the space we call our universe, the spacetime of general relativity, fractals, knots, manifolds, and phase spaces that occur in physics, as well as symmetry groups such as the collection of possible orientations of a spinning top. For example, the set of all possible positions of the hour hand of a clock is topologically equivalent to a circle (a one-dimensional closed curve with no self-intersections that can be embedded in two-dimensional space). The set of all possible positions of the hour and minute hands together is topologically equivalent to the surface of a torus (a two-dimensional surface that can be embedded in three-dimensional space). The set of all possible positions of the hour, minute, and second hands together is topologically equivalent to a three-dimensional object.

Topology is used in many branches of mathematics, such as differential equations, dynamical systems, knot theory, and Riemann surfaces in complex analysis. It is also used in string theory in physics and to describe the spacetime structure of the universe. This book deals with constructions such as subspaces, product spaces, and quotient spaces, as well as properties such as compactness and connectedness.

The edition is organized into seven chapters. Content has been revised on a chapter-by-chapter basis, and a separate chapter has been added. This is an introductory text in topology, or the study of shape. It covers point set topology, topological spaces and metric spaces, continuity and compactness, homotopy and covering spaces.

Idézet:? Topology 2nd Edition.pdf (GAMELIN, THEODORE W) (1999) (17.31 MB)