Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) *Give appropriate graphs and required codes. * Make use of inequalities if you think that required. *You are supposed to use kreszig for reference. (1.2) Definition: A space X is said to satisfy the T-axiom or is said to be a Ti-space if for every two distinct points x and y = X, there exists an open set containing x but not y (and hence also another open set contain- ing y but not x). Again, all metric spaces are T₁. It is obvious that every T₁ space is also To and the space (R, T) above shows that the converse is false. Thus the T₁-axiom is strictly stronger than To. (Sometimes a beginner fails to see any difference between the two conditions. The essential point is that given two distinct points, the To-axiom merely requires that at least one of them can be separated from the other by an open set whereas the T₁-axiom re- quires that each one of them can be separated from the other.) The following proposition characterises T1-spaces. (1.3) Proposition: For a topological space (X, T) the following are equivalent: (1) The space X is a T₁-space. (2) For any xX, the singleton set {x} is closed. (3) Every finite subset of X is closed. (4) The topology I is stronger than the cofinite topology on X. 3. The Tietze Extension Theorem • Problem: Prove the Tietze Extension Theorem, which states that if A is a closed subset of a normal topological space X and f: A→ [-1, 1] is continuous, then there exists a continuous extension F: X-1,1] such that F₁ = f • Details: ⚫ Define normal spaces, closed subsets, and continuous functions rigorously. • Use Urysohn's lemma and sequences of continuous functions to construct the extension step-by-step. Prove that the limit of these extensions remains within the range [-1,1]. ⚫ Graph: Illustrate the closed subset A and the continuous function f, with an extended function F defined over X.
Instructions: *Do not Use AI. (Solve by yourself, hand written preferred) *Give appropriate graphs and required codes. * Make use of inequalities if you think that required. *You are supposed to use kreszig for reference. (1.2) Definition: A space X is said to satisfy the T-axiom or is said to be a Ti-space if for every two distinct points x and y = X, there exists an open set containing x but not y (and hence also another open set contain- ing y but not x). Again, all metric spaces are T₁. It is obvious that every T₁ space is also To and the space (R, T) above shows that the converse is false. Thus the T₁-axiom is strictly stronger than To. (Sometimes a beginner fails to see any difference between the two conditions. The essential point is that given two distinct points, the To-axiom merely requires that at least one of them can be separated from the other by an open set whereas the T₁-axiom re- quires that each one of them can be separated from the other.) The following proposition characterises T1-spaces. (1.3) Proposition: For a topological space (X, T) the following are equivalent: (1) The space X is a T₁-space. (2) For any xX, the singleton set {x} is closed. (3) Every finite subset of X is closed. (4) The topology I is stronger than the cofinite topology on X. 3. The Tietze Extension Theorem • Problem: Prove the Tietze Extension Theorem, which states that if A is a closed subset of a normal topological space X and f: A→ [-1, 1] is continuous, then there exists a continuous extension F: X-1,1] such that F₁ = f • Details: ⚫ Define normal spaces, closed subsets, and continuous functions rigorously. • Use Urysohn's lemma and sequences of continuous functions to construct the extension step-by-step. Prove that the limit of these extensions remains within the range [-1,1]. ⚫ Graph: Illustrate the closed subset A and the continuous function f, with an extended function F defined over X.
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter5: Linear Inequalities
Section: Chapter Questions
Problem 2SGR
Related questions
Question
![Instructions:
*Do not Use AI. (Solve by yourself, hand written preferred)
*Give appropriate graphs and required codes.
* Make use of inequalities if you think that required.
*You are supposed to use kreszig for reference.
(1.2) Definition: A space X is said to satisfy the T-axiom or is said to
be a Ti-space if for every two distinct points x and y = X, there exists an
open set containing x but not y (and hence also another open set contain-
ing y but not x).
Again, all metric spaces are T₁. It is obvious that every T₁ space is also
To and the space (R, T) above shows that the converse is false. Thus the
T₁-axiom is strictly stronger than To. (Sometimes a beginner fails to see
any difference between the two conditions. The essential point is that given
two distinct points, the To-axiom merely requires that at least one of them
can be separated from the other by an open set whereas the T₁-axiom re-
quires that each one of them can be separated from the other.) The following
proposition characterises T1-spaces.
(1.3) Proposition: For a topological space (X, T) the following are
equivalent:
(1) The space X is a T₁-space.
(2) For any xX, the singleton set {x} is closed.
(3) Every finite subset of X is closed.
(4) The topology I is stronger than the cofinite topology on X.
3. The Tietze Extension Theorem
• Problem: Prove the Tietze Extension Theorem, which states that if A is a closed subset of a
normal topological space X and f: A→ [-1, 1] is continuous, then there exists a continuous
extension F: X-1,1] such that F₁ = f
• Details:
⚫ Define normal spaces, closed subsets, and continuous functions rigorously.
• Use Urysohn's lemma and sequences of continuous functions to construct the extension
step-by-step.
Prove that the limit of these extensions remains within the range [-1,1].
⚫ Graph: Illustrate the closed subset A and the continuous function f, with an extended
function F defined over X.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F495f0fd0-ab38-41cc-aa1b-3e24f7167a80%2F42be184c-7f2c-4b43-bfba-f271b83572a2%2Fgdwh6la_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Instructions:
*Do not Use AI. (Solve by yourself, hand written preferred)
*Give appropriate graphs and required codes.
* Make use of inequalities if you think that required.
*You are supposed to use kreszig for reference.
(1.2) Definition: A space X is said to satisfy the T-axiom or is said to
be a Ti-space if for every two distinct points x and y = X, there exists an
open set containing x but not y (and hence also another open set contain-
ing y but not x).
Again, all metric spaces are T₁. It is obvious that every T₁ space is also
To and the space (R, T) above shows that the converse is false. Thus the
T₁-axiom is strictly stronger than To. (Sometimes a beginner fails to see
any difference between the two conditions. The essential point is that given
two distinct points, the To-axiom merely requires that at least one of them
can be separated from the other by an open set whereas the T₁-axiom re-
quires that each one of them can be separated from the other.) The following
proposition characterises T1-spaces.
(1.3) Proposition: For a topological space (X, T) the following are
equivalent:
(1) The space X is a T₁-space.
(2) For any xX, the singleton set {x} is closed.
(3) Every finite subset of X is closed.
(4) The topology I is stronger than the cofinite topology on X.
3. The Tietze Extension Theorem
• Problem: Prove the Tietze Extension Theorem, which states that if A is a closed subset of a
normal topological space X and f: A→ [-1, 1] is continuous, then there exists a continuous
extension F: X-1,1] such that F₁ = f
• Details:
⚫ Define normal spaces, closed subsets, and continuous functions rigorously.
• Use Urysohn's lemma and sequences of continuous functions to construct the extension
step-by-step.
Prove that the limit of these extensions remains within the range [-1,1].
⚫ Graph: Illustrate the closed subset A and the continuous function f, with an extended
function F defined over X.
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