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 characterises 71-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. 12. Path Connectedness and Fundamental Groups •Problem: Prove that a path-connected space X has a fundamental group that is path- independent, meaning that the fundamental group (X,) is isomorphic to (X,21) for any two points 20, 21 X. • Details: ⚫ Define path-connectedness and the fundamental group at a base point. • Use homotopy and path concatenation to demonstrate the isomorphism between fundamental groups based on different base points. Graph: Show a path-connected space, such as a circle, with paths linking different base points, illustrating that loops based at different points can still be homotopic.