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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 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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 \(\mathbb{R}\) with the topology in which \(O \subset \mathbb{R}\) is open if and only if for each \(x \in O\), there is \(r > x\) with \([x, r) \subset O\). This is easily checked to be a topology, called the Sorgenfrey line topology. 1. Given \(x < r\) in \(\mathbb{R}\), - (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 \(\mathbb{R}\) is changed from the usual topology of \(\mathbb{R}\) to the Sorgenfrey line topology? Explain. 4. Explain why the Sorgenfrey line is not homeomorphic to the real line with its usual topology.
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 \(\mathbb{R}\) with the topology in which \(O \subset \mathbb{R}\) is open if and only if for each \(x \in O\), there is \(r > x\) with \([x, r) \subset O\). This is easily checked to be a topology, called the Sorgenfrey line topology. 1. Given \(x < r\) in \(\mathbb{R}\), - (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 \(\mathbb{R}\) is changed from the usual topology of \(\mathbb{R}\) 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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