Instructions: *Do not Use Al. (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 T₁. 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. 8. Metrization Theorem for Second Countable Spaces • Problem: Prove that any second-countable, regular, Hausdorff space is metrizable (i.e., there exists a metric that generates its topology). • Details: Define the conditions of second countability, regularity, and the Hausdorff property. • Construct a metric step-by-step that satisfies the topology, using a countable basis and separation properties. Graph: Illustrate this theorem by showing a second-countable topological space and a sample metric to depict convergence.
Instructions: *Do not Use Al. (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 T₁. 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. 8. Metrization Theorem for Second Countable Spaces • Problem: Prove that any second-countable, regular, Hausdorff space is metrizable (i.e., there exists a metric that generates its topology). • Details: Define the conditions of second countability, regularity, and the Hausdorff property. • Construct a metric step-by-step that satisfies the topology, using a countable basis and separation properties. Graph: Illustrate this theorem by showing a second-countable topological space and a sample metric to depict convergence.
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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Transcribed Image Text:Instructions:
*Do not Use Al. (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 T₁. 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.
8. Metrization Theorem for Second Countable Spaces
• Problem: Prove that any second-countable, regular, Hausdorff space is metrizable (i.e., there
exists a metric that generates its topology).
• Details:
Define the conditions of second countability, regularity, and the Hausdorff property.
• Construct a metric step-by-step that satisfies the topology, using a countable basis and
separation properties.
Graph: Illustrate this theorem by showing a second-countable topological space and a
sample metric to depict convergence.
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