# Are ordinal spaces extremally disconnected?

The wikipedia article on ordinal spaces claims that they are not extremally disconnected:

However, they are not extremally disconnected in general (there is an open set, namely $\omega$, whose closure is not open).

The justification seems wrong to me, since the closure of $\omega$ is $\omega+1$, which is open.

I wondered whether the Boolean algebras corresponding to the set algebras of clopen sets of the ordinal spaces (which are Stone spaces successor ordinals) are complete Boolean algebras, which should be equivalent to the question from the title.

• On the other hand, $\:\{2,\hspace{-0.03 in}4,\hspace{-0.03 in}6,\hspace{-0.03 in}8,\hspace{-0.04 in}10,...\hspace{-0.04 in}\}\:$ works to show that they are usually not extremally disconnected. $\hspace{.6 in}$
– user57159
Sep 21 '13 at 23:01

However, they are not extremally disconnected in general (there are open sets, for example the even numbers from $\omega$, whose closure is not open).