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.
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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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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