Exercise 1-4: Prove that the collection of all open half planes is a subbasis for the Euclidean topology of the plane.

Proof 1-4:

Note: In the Euclidean plane, we can take as a basis the collection of all interiors of squares.

Note also that a subcollection of all open sets of a topological space S is a SUBBASIS of S provided that the collection of all finite intersections of elements of $\mathcal{B}$ is a basis for S.

Clearly, from all open half planes ( and ), we can create a collection of all interior of squares.

Hence, the collection of all finite intersections of all open-half planes satisfies:

Axiom : the intersection of a finite intersection of open half planes is an open set. (interior of a square).

Axiom ; (also). The union of any number of finite intersections of all open half planes is also open set (interior of a square).

Axiom : (clearly). and S are open.

QED.

Exercise 1-5:

Let S be any infinite set. Show that requiring every infinite subset of S to be open imposes the discrete topology on S.

Proof 1-5:

Case: S is countable. We neglect the subcase that a selected subcase is finite. (I am using Prof. Rudin’s definition of countable). The other subcase is that there exists a subset , where is countable. Let be open. Hence, is closed. But is also countable. Hence, is also open. Hence, there is no limit point. Hence, the topology is discrete, that is, there is no limit point.

Case: S is uncountable. Consider again a proper subset ; hence, is open by imposition of hypothesis. Hence, is closed. But, and not finite also. Hence, is infinite. Hence, is open. Hence, there are no limit points. Hence, it is a discrete topology in this case also.

QED.

Your comments/observations are welcome !

Regards,

Nalin Pithwa.