This paper represents a continuation of our programme [16, 13] of extending various concepts of general topology from the setting of Hausdorff (or, at most, 7̂ ) spaces, in which they are usually embedded, to the larger classes of spaces we need to consider in the theory of computation. The topic of compactification poses an obvious challenge to this programme, since only a Tychonoff space can h...

The main purpose of this paper is to establish a relation between universality of certain P-compactifications of a semitopological semigroup and their corresponding enveloping semigroups. In particular, we show that if we take P to be the property that the enveloping semigroup of a compactification of a semitopological semigroup s is left simple, a group, or the trivial singleton semigroup, t...

Let X be a h-homogeneous zero-dimensional compact Hausdorff space, i.e. X is a Stone dual of a homogeneous Boolean algebra. It is shown that the universal minimal space M(G) of the topological group G = Homeo(X), is the space of maximal chains on X introduced in [Usp00]. If X is metrizable then clearly X is homeomorphic to the Cantor set and the result was already known (see [GW03]). However ma...

We exhibit Pontryagin duality as a special case of Stone in continuous logic setting. More specifically, given an abelian topological group A, and F the family (group) homomorphisms from A to circle T, then, viewing (A,+) equipped with collection structure M, we show that local type space SF(M) is precisely dual where latter considered discrete group. conclude, using (between compact groups), B...

If p : E → B is a continuous surjection between completely regular spaces E and B, we may apply the Stone-Čech compactification functor β to obtain a surjection βp : βE → βB. It is well-known that if E = B × F where F is a finite set and p is projection on the first factor, then βE = βB × βF , and βp is again projection on the first factor. In this paper, we apply β to an n-fold covering map, t...

In this note we obtain necessary and sufficient conditions for a convergence space to have a smallest Hausdorff compactification and to have a smallest regular compactification. Introduction. A Hausdorff convergence space as defined in [1] always has a Stone-Cech compactification which can be obtained by a slight modification of the result in [3]. But in general this need not be the largest Hau...

Having as starting point Barr’s description of topological spaces as lax algebras for the ultrafilter monad [2], in this paper we present further topological examples of lax algebras – such as quasi-metric spaces, approach spaces and quasi-uniform spaces – and show that, in a suitable setting, the categories of lax algebras have indeed a topological nature. Furthermore, we generalize to this se...