Extremally disconnected spaces are spaces, in which the closure of every open set is open. In this thesis we examine some interesting properties of extremally disconnec- ted Tychonoff spaces. We prove that in such spaces, every convergent sequence is constant from some point on. We show that a Tychonoff space is extremally disco- nected if and only if every subset of the partially ordered set C(X) that has an upper bound, also has a least upper bound. We also prove that an extremally disconnected Tychonoff topological field is discrete and that an extremally disconnected Tycho- noff topological group G is discrete if and only if the product G × G is extremally disconnected. In the end of this thesis we define Stone-Cech compactification in order to find an example of the spaces that we are studying.
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