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 T₁. It is obvious that every T₁ space is also To and the space (R, T) above shows that the converse is false. Thus the Ti-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 T₁-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. 5. The Baire Category Theorem and Applications ⚫ Problem: Prove the Baire Category Theorem: in a complete metric space, the intersection of countably many dense open sets is dense. Use this to show that Q is meager in R. ⚫ Details: Start with a clear definition of a complete metric space, open sets, dense sets, and meagerness. • Rigorously prove that the countable intersection of dense open sets remains dense. Apply this theorem to the rationals, showing it is of the first category in R. ⚫ Graph: Depict the density of open sets within a metric space, with an illustrative sequence converging to a point in the intersection.
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 T₁. It is obvious that every T₁ space is also To and the space (R, T) above shows that the converse is false. Thus the Ti-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 T₁-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. 5. The Baire Category Theorem and Applications ⚫ Problem: Prove the Baire Category Theorem: in a complete metric space, the intersection of countably many dense open sets is dense. Use this to show that Q is meager in R. ⚫ Details: Start with a clear definition of a complete metric space, open sets, dense sets, and meagerness. • Rigorously prove that the countable intersection of dense open sets remains dense. Apply this theorem to the rationals, showing it is of the first category in R. ⚫ Graph: Depict the density of open sets within a metric space, with an illustrative sequence converging to a point in the intersection.
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 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 T₁. It is obvious that every T₁ space is also
To and the space (R, T) above shows that the converse is false. Thus the
Ti-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 T₁-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.
5. The Baire Category Theorem and Applications
⚫ Problem: Prove the Baire Category Theorem: in a complete metric space, the intersection of
countably many dense open sets is dense. Use this to show that Q is meager in R.
⚫ Details:
Start with a clear definition of a complete metric space, open sets, dense sets, and
meagerness.
• Rigorously prove that the countable intersection of dense open sets remains dense.
Apply this theorem to the rationals, showing it is of the first category in R.
⚫ Graph: Depict the density of open sets within a metric space, with an illustrative sequence
converging to a point in the intersection.
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