Explain

26. Show that the product topology on a product set X = 1, X, is coarser than the topology T on X defined in the preceding problem (Proposition 12.8). 27. Give an example of a topology 7 on a product set X = II, X, which is coarser than the product topology on X. 28. Let 7 be the topology on a product set X = II, X, defined in Problem 25 (Proposition 12.8). Show that (X, T) is discrete if each coordinate space X, is discrete. FINITE PRODUCTS 29. Prove Proposition 12.2: The subsets of a product space X = X, X . .. XX, of the form G, X . . . X G, where G, is an open subset of X, form a base for the product topology on X. 30. Prove: If B is a base for X and B* is a base for Y, then (G X H : GE B, HEB*) is a base for the product space X X Y. 31. Prove: If Ba is a local base at a E X and B, is a local base at bE Y, then (G X H : GE BQ, HEB) is a local base at p = (a, b) e XX Y. 32. Prove that the product of two first countable spaces is first countable. 33. Prove that the product of two second countable spaces is second countable. 34. Prove that the product of two separable spaces is separable. 35. Prove that the product of two compact spaces is compact (without using Zorn's Lemma or its equivalents). 36. Let B" be the topology on the plane R2 generated by the half-open rectangles [a, b) X [c, d) = {(x,y) : a = x