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 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…

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 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 7₁. 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…

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 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 7₁. 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…

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…

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…

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 T-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 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…

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…

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 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…

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 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 T-axiom re- quires that each one of them can be separated from the other.) The following proposition…