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 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, 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 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. 2. Connectedness in Topological Spaces • Problem: Show that the continuous image of a connected space is connected. Furthermore, prove that the real interval [a, b] is connected in the standard topology of R. ⚫ Details: • Begin with the definitions of connectedness and continuous maps. • Use the intermediate value theorem and properties of continuous functions to rigorously demonstrate the preservation of connectedness. Prove that [a, b] is connected by assuming the contrary and deriving a contradiction. • Graph: Draw an interval [a, b] with no separation, and an example of a continuous mapping to a connected subset of R.
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 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, 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 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. 2. Connectedness in Topological Spaces • Problem: Show that the continuous image of a connected space is connected. Furthermore, prove that the real interval [a, b] is connected in the standard topology of R. ⚫ Details: • Begin with the definitions of connectedness and continuous maps. • Use the intermediate value theorem and properties of continuous functions to rigorously demonstrate the preservation of connectedness. Prove that [a, b] is connected by assuming the contrary and deriving a contradiction. • Graph: Draw an interval [a, b] with no separation, and an example of a continuous mapping to a connected subset of R.
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 T₁-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, 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 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.
2. Connectedness in Topological Spaces
• Problem: Show that the continuous image of a connected space is connected. Furthermore,
prove that the real interval [a, b] is connected in the standard topology of R.
⚫ Details:
• Begin with the definitions of connectedness and continuous maps.
• Use the intermediate value theorem and properties of continuous functions to rigorously
demonstrate the preservation of connectedness.
Prove that [a, b] is connected by assuming the contrary and deriving a contradiction.
• Graph: Draw an interval [a, b] with no separation, and an example of a continuous mapping
to a connected subset of R.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90bddbb1-7b43-41c9-9287-454ecc97e588%2F01974799-15c9-480c-86cc-2596a999d9ad%2Fob3d6yf_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 T₁-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, 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 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.
2. Connectedness in Topological Spaces
• Problem: Show that the continuous image of a connected space is connected. Furthermore,
prove that the real interval [a, b] is connected in the standard topology of R.
⚫ Details:
• Begin with the definitions of connectedness and continuous maps.
• Use the intermediate value theorem and properties of continuous functions to rigorously
demonstrate the preservation of connectedness.
Prove that [a, b] is connected by assuming the contrary and deriving a contradiction.
• Graph: Draw an interval [a, b] with no separation, and an example of a continuous mapping
to a connected subset of R.
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