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 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 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 Ti-axiom re- quires that each one of them can be separated from the other.) The following proposition characterises Ti-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. 13. Homotopy Lifting Property in Covering Spaces Problem: Prove that if p: X→ X is a covering map, then it has the homotopy lifting property: for any map f: YX and homotopy F : Y x [0,1] → X, there exists a lift F: Y x [0,1] → X such that po F = F. • Details: Define the covering space, covering map, and homotopy lifting property. • Use path lifting arguments and properties of covering spaces to construct a lifting homotopy F • Graph: Draw a covering space й and its base space X, with a map lifting a path in X to X ,illustrating the homotopy lifting property.
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 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 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 Ti-axiom re- quires that each one of them can be separated from the other.) The following proposition characterises Ti-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. 13. Homotopy Lifting Property in Covering Spaces Problem: Prove that if p: X→ X is a covering map, then it has the homotopy lifting property: for any map f: YX and homotopy F : Y x [0,1] → X, there exists a lift F: Y x [0,1] → X such that po F = F. • Details: Define the covering space, covering map, and homotopy lifting property. • Use path lifting arguments and properties of covering spaces to construct a lifting homotopy F • Graph: Draw a covering space й and its base space X, with a map lifting a path in X to X ,illustrating the homotopy lifting property.
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter5: Linear Inequalities
Section: Chapter Questions
Problem 2SGR
Related questions
Question
![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 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
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 Ti-axiom re-
quires that each one of them can be separated from the other.) The following
proposition characterises Ti-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.
13. Homotopy Lifting Property in Covering Spaces
Problem: Prove that if p: X→ X is a covering map, then it has the homotopy lifting property:
for any map f: YX and homotopy F : Y x [0,1] → X, there exists a lift F: Y x
[0,1] → X such that po F = F.
• Details:
Define the covering space, covering map, and homotopy lifting property.
• Use path lifting arguments and properties of covering spaces to construct a lifting
homotopy F
• Graph: Draw a covering space й and its base space X, with a map lifting a path in X to X
,illustrating the homotopy lifting property.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F816bf715-adf0-4442-85f5-3a7030a36b31%2F869f4a88-1165-4d84-8ac0-c824790ca42a%2Fdh9y6a_processed.jpeg&w=3840&q=75)
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 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
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 Ti-axiom re-
quires that each one of them can be separated from the other.) The following
proposition characterises Ti-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.
13. Homotopy Lifting Property in Covering Spaces
Problem: Prove that if p: X→ X is a covering map, then it has the homotopy lifting property:
for any map f: YX and homotopy F : Y x [0,1] → X, there exists a lift F: Y x
[0,1] → X such that po F = F.
• Details:
Define the covering space, covering map, and homotopy lifting property.
• Use path lifting arguments and properties of covering spaces to construct a lifting
homotopy F
• Graph: Draw a covering space й and its base space X, with a map lifting a path in X to X
,illustrating the homotopy lifting property.
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