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Quoted from an email originally sent to: unstattedCommoner

**The author writes:**

Let's see if I can explain this:

A topological space (A, T) is a set A together with a collection T of subsets of A, where T satisfies the following:

- The whole set A and the empty set are in T;
- The intersection of any two sets in T is in T;
- The union of any collection of sets in T is in T.

T is said to be a topology for A. The members of T are called "open sets". Note that not all subsets of A need be in T. Given x in A, an open set which contains x is said to be a "neighbourhood" of x.

Given A, one has the "discrete topology" in which {x} is open for all x in A; in this case, all subsets of A are open. There is also the "indiscrete topology" in which the only open sets are A and the empty set.

A topology T1 is said to be "coarser" than a topology T2 if T1 is a subcollection of T2; T2 is then said to be "finer" than T1. It should be obvious that the discrete topology is the finest and the indiscrete topology the most coarse.

Hence the puns.