Let be a set.

Let .

Then the excluded point topology is given by:

A subset of is open iff

The argument given in Counterexamples in Topology by Steen and Seebach (counterexamples 13-15: 5) goes:

Let . Then is open in as .

So by definition can not be a limit point and so is an isolated point.

So any subset is not dense-in-itself, because it contains isolated points.

So by definition is scattered.

There's a flaw in the above, in that is easily shown to be a limit point (the only one possible in , in fact) and so has no isolated points in it, so is dense-in-itself.

So it appears that is not scattered after all, as it contains (exactly) one dense-in-itself subset.

Is Steen and Seebach wrong? Or have I missed something, e.g. " is required to be open to be described as dense-in-itself" or something? But I can't find any such words.