General Topology I Basic Concepts and Constructions Dimension Theory

General topology is the domain of mathematics devoted to the investigation of the concepts of continuity and passage to a limit at their natural level of generality. The most basic concepts of general topology, that of a topological space and a continuous map, were introduced by Hausdorff in 1914. One ofthe central problems oftopology is the determination and investigation of topological invariants; that is, properties of spaces which are preserved under homeomorphisms.
Topological invariants need not be numbers. Connectedness, compactness, and metrizability, for example, are non-numerical topological invariants. Dimensional invariants, on the other hand, are examples of numerical invariants which take integer values on specific topological spaces. Part II of this book is devoted to them. Topological invariants which take values in the cardinal numbers play an especially important role, providing the raw material for many useful coin" putations. Weight, density, character, and Suslin number are invariants of this type.
Certain classes of topological spaces are defined in terms of topological invariants. Particularly important examples include the metrizable spaces, spaces with a countable base, compact spaces, Tikhonov spaces, Polish spaces, Cechcomplete spaces and the symmetrizable spaces.