View source for 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|perfect irreducible mapping]] of some extremally-disconnected Tikhonov space (see [[Absolute|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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037240/e0372401.png" />-space is discrete.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>



====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|Boolean algebra]]) extremally-disconnected compacta correspond to complex Boolean algebras.

For a discussion of homogeneity see [[Homogeneous space|Homogeneous space]].