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 T₁-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 71. 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 T-axiom re- quires that each one of them can be separated from the other.) The following proposition characterises Ti-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. 9. Jordan Curve Theorem • Problem: Prove the Jordan Curve Theorem, which states that a simple closed curve in R² divides the plane into an "inside" and an "outside," forming two connected components. ⚫ Details: ⚫ Define a simple closed curve and connected components rigorously. • Use the concept of continuity, connectedness, and path-connectedness to outline the proof. • Graph: Draw a simple closed curve (like a circle or an irregular closed loop) and label the interior and exterior regions, indicating that each point on the curve belongs to one of these connected components.
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 T₁-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 71. 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 T-axiom re- quires that each one of them can be separated from the other.) The following proposition characterises Ti-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. 9. Jordan Curve Theorem • Problem: Prove the Jordan Curve Theorem, which states that a simple closed curve in R² divides the plane into an "inside" and an "outside," forming two connected components. ⚫ Details: ⚫ Define a simple closed curve and connected components rigorously. • Use the concept of continuity, connectedness, and path-connectedness to outline the proof. • Graph: Draw a simple closed curve (like a circle or an irregular closed loop) and label the interior and exterior regions, indicating that each point on the curve belongs to one of these connected components.
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
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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 T₁-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 71. 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 T-axiom re-
quires that each one of them can be separated from the other.) The following
proposition characterises Ti-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.
9. Jordan Curve Theorem
• Problem: Prove the Jordan Curve Theorem, which states that a simple closed curve in R² divides
the plane into an "inside" and an "outside," forming two connected components.
⚫ Details:
⚫ Define a simple closed curve and connected components rigorously.
• Use the concept of continuity, connectedness, and path-connectedness to outline the
proof.
• Graph: Draw a simple closed curve (like a circle or an irregular closed loop) and label the
interior and exterior regions, indicating that each point on the curve belongs to one of
these connected components.
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