Section 13: Problem 5 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Show that if $\mathcal{A}$ is a basis for a topology on $X$ , then the topology generated by $\mathcal{A}$ equals the intersection of all topologies on $X$ that contain $\mathcal{A}$ . Prove the same if $\mathcal{A}$ is a subbasis.

Every topology containing the collection $\mathcal{A}$ must contain all unions of sets of $\mathcal{A}$ , i.e. it must contain the topology generated by $\mathcal{A}$ .

If $\mathcal{A}$ is a subbasis, then every topology containing $\mathcal{A}$ must contain all finite intersections of sets of $\mathcal{A}$ , i.e. it must contain the basis generated by the subbasis $\mathcal{A}$ .

In both cases, the topology generated by $\mathcal{A}$ contains $\mathcal{A}$ , but at the same time is contained in every topology that contains $\mathcal{A}$ , hence, it equals the intersection of such topologies (which is the smallest topology containing $\mathcal{A}$ ).

Subpages

Section 13: Problem 5 Solution

Section 13: Problem 5 Solution