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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![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 O C R is open if and only if for each x € O, there is r > x
with [x, r) C O. 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c557725-037f-4839-bada-f183764fca9b%2Fea06c0d3-3f4e-4e3e-b974-db1e46e43a93%2F7c4fdo_processed.png&w=3840&q=75)
Transcribed Image Text: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 O C R is open if and only if for each x € O, there is r > x
with [x, r) C O. 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.
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