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 T₁-space if for every two distinct points x and y E 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, T) 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 T₁-axiom re- quires that each one of them can be separated from the other.) The following proposition characterises T-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. 11. Boundary of Open Sets in Metric Spaces • Problem: Prove that in a metric space, the boundary of an open set is closed and has empty interior. ⚫ Details: • Define open and closed sets and their boundaries. • Use sequences or limiting arguments to show that the boundary of an open set is closed. • Prove that any open neighborhood around a boundary point will intersect both the set and its complement, ensuring empty interior. • Graph: Draw an open set in a metric space and mark its boundary, showing the absence of any interior points within this boundary.
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 T₁-space if for every two distinct points x and y E 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, T) 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 T₁-axiom re- quires that each one of them can be separated from the other.) The following proposition characterises T-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. 11. Boundary of Open Sets in Metric Spaces • Problem: Prove that in a metric space, the boundary of an open set is closed and has empty interior. ⚫ Details: • Define open and closed sets and their boundaries. • Use sequences or limiting arguments to show that the boundary of an open set is closed. • Prove that any open neighborhood around a boundary point will intersect both the set and its complement, ensuring empty interior. • Graph: Draw an open set in a metric space and mark its boundary, showing the absence of any interior points within this boundary.
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 T-axiom or is said to
be a T₁-space if for every two distinct points x and y E 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, T) 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 T₁-axiom re-
quires that each one of them can be separated from the other.) The following
proposition characterises T-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.
11. Boundary of Open Sets in Metric Spaces
• Problem: Prove that in a metric space, the boundary of an open set is closed and has empty
interior.
⚫ Details:
• Define open and closed sets and their boundaries.
• Use sequences or limiting arguments to show that the boundary of an open set is closed.
• Prove that any open neighborhood around a boundary point will intersect both the set and
its complement, ensuring empty interior.
• Graph: Draw an open set in a metric space and mark its boundary, showing the absence of
any interior points within this boundary.
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