Reference: Introductory Real Analysis by Kolmogorov and Fomin, Dover Pub.
Reference: Analysis by Walter Rudin
Section 5.2: Continuous mappings and homeomorphisms. isometric spaces:
Let f be a mapping of one metric space X into another metric space Y, so that f associates an element with each element . Then, f is said to be continuous at the point if, given any , there exists a such that
In the above, the metric is in X and the metric is in Y. The mapping f is said to be continuous on X if it is continuous at every point in domain X.
This definition reduces to the usual definition of continuity familiar from calculus if X and Y are both numerical sets, that is, if f is a real function defined on some subset of this real line.
Given two metric spaces X and Y, let f be one-to-one mapping of X onto X, and suppose and are both continuous. Then, f is called a homeomorphic mapping, or simply a homeomorphism between X and Y. Two spaces X and Y are said to be homeomorphic if there exists a homeomorphism between them.
are said to be isometric to each other. The function
establishes a homeomorphism between the whole real line and the open interval .
A one-to-one mapping f of one metric space onto another metric space is said to be an isometric mapping (or simply, an isometry) if
for all . Correspondingly, the spaces R and are said to be isometric to each other.
Thus, if and are isometric, the “metric relations” between the elements of R are the same as those between the elements of , that is, R and differ only in the explicit nature of their elements (this distinction is unimportant from the viewpoint of metric space theory). From now on, we will not distinguish between isometric spaces regarding them simply as identical.
We will discuss continuity and homeomorphism from a more general viewpoint later.
The next blog poses some exercise problems.