C-1 - Continuous and Topological Mappings

Abstract

This chapter discusses the concepts of continuous and topological mappings. It is assumed that X, Y, and Z denote topological spaces. If ƒ: X→Y is a map from X into Y, x is a point of X, and for every open neighborhood V of ƒ(x) in Y there is an open neighborhood U of x such that ƒ(U) ⊂ V, then the map ƒ is said to be continuous at x. If ƒ is continuous at every point of X, then it is called a continuous map, in other words ƒ is continuous on X. A continuous map from X into ℝ (ℂ) is often called a real-valued (complex-valued) continuous function on X. Real-valued continuous functions of one real variable are the best known examples of continuous maps (from ℝ into ℝ ). Another example illustrates that every map from a discrete space into any space is a continuous map. Perhaps the most important type of continuous map is a topological map, whose definition is given as: If ƒ: X →Y is a continuous bijection (one-to-one onto map) from X onto Y, and if the inverse map ƒ−1 is also continuous, then ƒ is called a topological map or homeomorphism, and X and Y are said to be homeomorphic spaces.