chapter  2
44 Pages

Topological Spaces

WithL. S. Pontryagin

This chapter provides a means of defining a topological space in terms of neighborhoods. This method is particularly important and is, indeed, frequently made the basis of the definition of a topological space. Making use of the fact that the union of an arbitrary collection of open sets is open, people are led to a further simplification. The conditions and just formulated are important in that they may in their turn be taken as the axioms in a definition of a topological space by means of neighborhoods. A more precise formulation of this idea is given in Theorem 3, which at the same time provides a converse and taken together. From the topological point of view two topological spaces whose closure operations have the same structure are indistinguishable; they are said to be homeomorphic. A connection between two spaces weaker than that of homeomorphism is given by a continuous mapping.