Exercise 11. Recall the following Example of a topological space with the real line as underlying set (Example 26 in the notes): Consider on the real line R with the topology in which OCR is open if and only if for each x EO, there is r > x with [x,r) CO. This is easily checked to be a topology, called the Sorgenfrey line topology. (1) Given x < r in R, (a) Is [x, r) open? Justify why or why not. (b) Is [x, r) closed? Justify why or why not. (2) Is every open subset of the Sorgenfrey line also closed? Justify why or why not. (3) Is the property of Exercise 4 still valid when the topology on R is changed from the usual topology of R to the Sorgenfrey line topology? Explain. (4) Explain why the Sorgenfrey line is not homeomorphic to the real line with its usual topology.

Exercise 11. Recall the following Example of a topological space with the real line as underlying set (Example 26 in the notes): Consider on the real line R with the topology in which OCR is open if and only if for each x EO, there is r > x with [x,r) CO. This is easily checked to be a topology, called the Sorgenfrey line topology. (1) Given x < r in R, (a) Is [x, r) open? Justify why or why not. (b) Is [x, r) closed? Justify why or why not. (2) Is every open subset of the Sorgenfrey line also closed? Justify why or why not. (3) Is the property of Exercise 4 still valid when the topology on R is changed from the usual topology of R to the Sorgenfrey line topology? Explain. (4) Explain why the Sorgenfrey line is not homeomorphic to the real line with its usual topology.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 6TFE: Label each of the following statements as either true or false. Let R be a relation on a nonempty...

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