Aspace is totally disconnected if its connected spaces are one-point-sets. show that a finite hausdorff space is totally disconnected.

Step-by-step explanation:

If X is a finite Hausdorff space then every two points of X can be separated by open neighborhoods. Say the points of X are . So there are disjoint open neighborhoods and , of and respectively (that's the definition of Hausdorff space). There are also open disjoint neighborhoods and of and respectively, and disjoint open neighborhoods and of and , and so on, all the way to disjoint open neighborhoods , and of and respectively. So has every element of in it, except for . Since is union of open sets, it is open, and so , which is the singleton , is closed. Therefore every singleton is closed.

Now, remember finite union of closed sets is closed, so is closed, and so its complemented, which is is open. Therefore every singleton is also open.

That means any two points of belong to different connected components (since we can express X as the union of the open sets , so that is in a different connected component than , and same could be done with any ), and so each point is in its own connected component. And so the space is totally disconnected.

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