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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 T-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, I) 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 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. 14. Alexander Duality in R³ • Problem: Prove Alexander Duality for compact subsets in Rª: for a compact subset K CR³, there is a relationship between the homology of K and the homology of its complement. ⚫ Details: • Define homology groups, compact subsets, and duality in the context of topological spaces. • Use Mayer-Vietoris sequences and excision to set up the proof, demonstrating the complementary homological relationship. ⚫Graph: Show a compact subset K (such as a sphere or torus) and its complement in R³, with illustrations for homology groups for both spaces.

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 T-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, I) 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 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. 14. Alexander Duality in R³ • Problem: Prove Alexander Duality for compact subsets in Rª: for a compact subset K CR³, there is a relationship between the homology of K and the homology of its complement. ⚫ Details: • Define homology groups, compact subsets, and duality in the context of topological spaces. • Use Mayer-Vietoris sequences and excision to set up the proof, demonstrating the complementary homological relationship. ⚫Graph: Show a compact subset K (such as a sphere or torus) and its complement in R³, with illustrations for homology groups for both spaces.

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter5: Linear Inequalities
Section: Chapter Questions
Problem 2SGR
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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 T-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, I) 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 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. 14. Alexander Duality in R³ • Problem: Prove Alexander Duality for compact subsets in Rª: for a compact subset K CR³, there is a relationship between the homology of K and the homology of its complement. ⚫ Details: • Define homology groups, compact subsets, and duality in the context of topological spaces. • Use Mayer-Vietoris sequences and excision to set up the proof, demonstrating the complementary homological relationship. ⚫Graph: Show a compact subset K (such as a sphere or torus) and its complement in R³, with illustrations for homology groups for both spaces.
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 T-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, I) 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 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. 14. Alexander Duality in R³ • Problem: Prove Alexander Duality for compact subsets in Rª: for a compact subset K CR³, there is a relationship between the homology of K and the homology of its complement. ⚫ Details: • Define homology groups, compact subsets, and duality in the context of topological spaces. • Use Mayer-Vietoris sequences and excision to set up the proof, demonstrating the complementary homological relationship. ⚫Graph: Show a compact subset K (such as a sphere or torus) and its complement in R³, with illustrations for homology groups for both spaces.
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