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I believe the following is true.

In a Hausdorff space, every one point set is closed. (For any other point than the point in question there is an open set not intersecting it, thus the single point is closed.)

An isolated point is defined as one point set being open.
Thus a Hausdorff space contains no isolated points.


Yet in proofs that compact Hausdorff spaces are uncountable it is also assumed that the space contains no isolated points.

Can a Hausdorff space contain single point sets that are open? Any example of an isolated point in a Hausdorff space?

Thanks in advance!

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    $\begingroup$ Take any (nonempty) set with the discrete topology. All points are isolated. A compact (Hausdorff) space has only finitely many isolated points, but the number need not be $0$. $\endgroup$ – Daniel Fischer Nov 1 '14 at 21:14
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    $\begingroup$ The key as others have pointed out is that open and closed are not mutually exclusive states. $\endgroup$ – Alan Nov 1 '14 at 21:17
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A set can be both open and closed. For example, the empty set and the whole space are both open and closed. If a Hausdorff space has has a finite number of points, then each point is both open and closed (open since it is the complement of a closed set, and closed because it is the complement of an open set).

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The discrete topology is Hausdorff, and every point is isolated.

Also, consider $[0,1] \cup \{2\}$ under the subspace topology inherited from $\Bbb R$; it is compact, Hausdorff, and $2$ is an isolated point.

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