Basic Topology the rational2. Let X be the reals and define p(x, y) = 2x - y). Show that p is a pseudo-metric equivalent to the usual pseudometric for X.3. Let X be the plane and for x = (1, 2) and y = (v₁.32), let p(x, y) =1-₁+12-y2|-(a) Show that p is a pseudometric.(b) Describe the p-cell of radius r centered at the point (a, b).(e) Find the p-closure of S = {re X: r + x² < 1}.

Answered: 2. Let X be the reals and define p(x,…

2. Let X be the reals and define p(x, y) = 2/x - y). Show that p is a pseudo- metric equivalent to the usual pseudometric for X.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....

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Question

Basic Topology

the rational
2. Let X be the reals and define p(x, y) = 2x - y). Show that p is a pseudo-
metric equivalent to the usual pseudometric for X.
3. Let X be the plane and for x = (1, 2) and y = (v₁.32), let p(x, y) =
1-₁+12-y2|-
(a) Show that p is a pseudometric.
(b) Describe the p-cell of radius r centered at the point (a, b).
(e) Find the p-closure of S = {re X: r + x² < 1}.

Expert Solution

Step 1

To show that the function d(x,y) = 2|x-y| is a pseudometric on X, we need to show that it satisfies the following properties:

Non-negativity: d(x,y) ≥ 0 for all x,y in X, and d(x,y) = 0 if and only if x = y.
Symmetry: d(x,y) = d(y,x) for all x,y in X.
Triangle inequality: d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z in X.

Note: we are taking d(x,y) instead of row(x,y) for convenience of writing the symbol

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