Extremally-disconnected space

A space in which the closure of every open set is open. In a regular extremally-disconnected space there are no convergent sequences without repeated terms. Therefore, among the metric spaces only the discrete ones are extremally disconnected. Nevertheless, extremally-disconnected spaces are fairly widespread: Every Tikhonov space can be represented as the image under a perfect irreducible mapping of some extremally-disconnected Tikhonov space (see Absolute of a topological space). This means that extremal disconnectedness is not preserved by perfect mappings. However, the image of an extremally-disconnected space under a continuous open mapping is an extremally-disconnected space.

All regular extremally-disconnected spaces are zero-dimensional; however, unlike zero-dimensionality, extremal disconnectedness is not inherited by arbitrary subspaces, not even by closed ones. But an everywhere-dense subspace of an extremally-disconnected space is always extremally disconnected. Extremal disconnectedness does not combine well with topological homogeneity. In particular, every extremally-disconnected topologically homogeneous compactum is finite. Nevertheless, under the continuum hypothesis there is a topological group which is a non-discrete extremally-disconnected Hausdorff space. Every compact subspace of an extremally-disconnected topological Hausdorff group is finite. Hence, every extremally-disconnected topological group whose space is a $k$-space is discrete.

References

[1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)

Comments

Instead of "convergent sequence without repeated terms" one also uses the phrase "non-trivial convergent sequencenon-trivial convergent sequence" .

Via Stone duality (see Boolean algebra) extremally-disconnected compacta correspond to complex Boolean algebras.

For a discussion of homogeneity see Homogeneous space.