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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 7-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 x = X, 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. 4. Classification of Surfaces Using Euler Characteristic • Problem: Prove that the Euler characteristic X classifies surfaces up to homeomorphism in the case of orientable and non-orientable surfaces. ⚫ Details: • Begin with definitions of surfaces, orientable/non-orientable, and Euler characteristic. Show that surfaces with the same Euler characteristic are homeomorphic by decomposing them into simpler components. • Prove that no two surfaces with different Euler characteristics are homeomorphic. ⚫ Graph: Provide examples of surfaces like the sphere, torus, and Möbius strip with calculated Euler characteristics to illustrate the classification.

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 7-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 x = X, 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. 4. Classification of Surfaces Using Euler Characteristic • Problem: Prove that the Euler characteristic X classifies surfaces up to homeomorphism in the case of orientable and non-orientable surfaces. ⚫ Details: • Begin with definitions of surfaces, orientable/non-orientable, and Euler characteristic. Show that surfaces with the same Euler characteristic are homeomorphic by decomposing them into simpler components. • Prove that no two surfaces with different Euler characteristics are homeomorphic. ⚫ Graph: Provide examples of surfaces like the sphere, torus, and Möbius strip with calculated Euler characteristics to illustrate the classification.

Mathematics For Machine Technology
Mathematics For Machine Technology
8th Edition
ISBN:9781337798310
Author:Peterson, John.
Publisher:Peterson, John.
Chapter87: An Introduction To G- And M-codes For Cnc Programming
Section: Chapter Questions
Problem 25A
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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 7-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 x = X, 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. 4. Classification of Surfaces Using Euler Characteristic • Problem: Prove that the Euler characteristic X classifies surfaces up to homeomorphism in the case of orientable and non-orientable surfaces. ⚫ Details: • Begin with definitions of surfaces, orientable/non-orientable, and Euler characteristic. Show that surfaces with the same Euler characteristic are homeomorphic by decomposing them into simpler components. • Prove that no two surfaces with different Euler characteristics are homeomorphic. ⚫ Graph: Provide examples of surfaces like the sphere, torus, and Möbius strip with calculated Euler characteristics to illustrate the classification.
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 7-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 x = X, 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. 4. Classification of Surfaces Using Euler Characteristic • Problem: Prove that the Euler characteristic X classifies surfaces up to homeomorphism in the case of orientable and non-orientable surfaces. ⚫ Details: • Begin with definitions of surfaces, orientable/non-orientable, and Euler characteristic. Show that surfaces with the same Euler characteristic are homeomorphic by decomposing them into simpler components. • Prove that no two surfaces with different Euler characteristics are homeomorphic. ⚫ Graph: Provide examples of surfaces like the sphere, torus, and Möbius strip with calculated Euler characteristics to illustrate the classification.
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