Metric Spaces: some additional stuff

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 $y=f(x) \in Y$ with each element $x \in X$. Then, f is said to be continuous at the point $x_{0}$ if, given any $\epsilon >0$, there exists a $\delta >0$ such that

$\rho^{'}(f(x),f(x_{0})) < \epsilon$ whenever $\rho(x,x_{0}) < \delta$.

In the above, the metric $\rho$ is in X and the metric $\rho^{'}$ is in Y. The mapping f is said to be continuous on X if it is continuous at every point in domain X.

Remarks:

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 $f$ and $f^{-1}$ 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.

Example :

are said to be isometric to each other. The function $y=f(x) = \frac{2}{\pi}\arctan{x}$

establishes a homeomorphism between the whole real line $(-\infty, \infty)$ and the open interval $(-1,1)$.

DEFINITION 2:

A one-to-one mapping f of one metric space $R=(X,\rho)$ onto another metric space $R^{'} = (X, \rho^{'})$ is said to be an isometric mapping (or simply, an isometry) if

$\rho(x_{1}, x_{2}) = \rho^{'}(f(x_{1}), f(x_{2}))$

for all $x_{1}, x_{2} \in R$. Correspondingly, the spaces R and $R^{'}$ are said to be isometric to each other.

Thus, if $R$ and $R^{'}$ are isometric, the “metric relations” between the elements of R are the same as those between the elements of $R^{'}$, that is, R and $R^{'}$ 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.

Remark:

We will discuss continuity and homeomorphism from a more general viewpoint later.

The next blog poses some exercise problems.

Regards,

Nalin Pithwa.

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