Fixed point compactifications

A fixed point compactification of a locally compact noncompact group G is a faithful semigroup compactification S such that \(ap=pa=p\) for all \(p\in S\setminus G\) and \(a\in G\). Since the right translations are continuous, the remainder of a fixed point compactification is a right zero semigroup. Among all fixed point compactifications of G there is a largest one, denoted \(\theta G\). We show that if G is \(\sigma \)-compact, then \(\theta G\setminus G\) contains a copy of \(\beta \omega \setminus \omega \). In contrast, if G is not \(\sigma \)-compact, then \(\theta G\) is the one-point compactification.     

Metrizable mapping compactifications

Suppose f:X→f(X)=Y is a continuous function from one completely regular Hausdorff space onto another. There is associated with each possible compactification X̃ of the domain space X a compactification of the mapping f in a unique way; the mapping compactification is called the compactification determined by X̃. The major result of this paper is that if X̃ is metrizable, then the domain of the mapping compactification determined by it is also metrizable if and only if the range Y is. It is also proved that if X₁ and X₂ are two compactifications of X such that X₂⩾X₁, where ⩾ is the usual partial order on the collection of all compactifications of X, then the compactifications of f determined by X₁ and X₂ are related the same way with respect to the usual partial order on the collection of all compactifications of f.     

Submaximal and door compactifications

In this paper, a characterization is given for compact door spaces. We, also, deal with spaces X such that a compactification K(X) of X is submaximal or door.Let X be a topological space and K(X) be a compactification of X.We prove, here, that K(X) is submaximal if and only if for each dense subset D of X, the following properties hold:

(i)

D is co-finite in K(X);

(ii)

for each x∈K(X)?D, {x} is closed.

If X is a noncompact space, then we show that K(X) is a door space if and only if X is a discrete space and K(X) is the one-point compactification of X.     

Function-space compactifications of function spaces

If X and Y are Hausdorff spaces with X locally compact, then the compact-open topology on the set C(X,Y) of continuous maps from X to Y is known to produce the right function-space topology. But it is also known to fail badly to be locally compact, even when Y is locally compact. We show that for any Tychonoff space Y, there is a densely injective space Z containing Y as a densely embedded subspace such that, for every locally compact space X, the set C(X,Z) has a compact Hausdorff topology whose relative topology on C(X,Y) is the compact-open topology. The following are derived as corollaries: (1) If X and Y are compact Hausdorff spaces then C(X,Y) under the compact-open topology is embedded into the Vietoris hyperspace V(X×Y). (2) The space of real-valued continuous functions on a locally compact Hausdorff space under the compact-open topology is embedded into a compact Hausdorff space whose points are pairs of extended real-valued functions, one lower and the other upper semicontinuous. The first application is generalized in two ways.     

Hausdorff compactifications and lebesgue sets

Let X be a completely regular Hausdorff space and let H be a subset of C¹(X) which separates points and closed sets. By embedding X into a cube whose factors are indexed by H, a Hausdorff compactification e_HX of X is obtained. Given two subsets F and G of C¹(X) which separate points from closed sets, in the present paper we obtain a necessary and sufficient condition for the equivalence of e_FX and e_GX. The condition is expressed in terms of the space X and the sets F and G alone, herewith solving a question raised by Chandler.     

Hausdorff compactifications and Wallman spaces

Acta Mathematica Hungarica -     

Manifold-theoretic compactifications of configuration spaces

We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to Fulton–MacPherson and Axelrod–Singer in the setting of smooth manifolds, as well as a simplicial variant of this compactification initiated by Kontsevich. Our constructions are elementary and give simple global coordinates for the compactified configuration space of a general manifold embedded in Euclidean space. We stratify the canonical compactification, identifying the difieomorphism types of the strata in terms of spaces of configurations in the tangent bundle, and give completely explicit local coordinates around the strata as needed to define a manifold with corners. We analyze the quotient map from the canonical to the simplicial compactification, showing it is a homotopy equivalence. Using global coordinates we define projection maps and diagonal maps, which for the simplicial variant satisfy cosimplicial identities.     

A characterization of fuzzy compactifications

It is shown that if X is a fuzzy T₂-space, then X has a fuzzy T₂-compactification if and only if X is a weakly induced ultra completely regular space. Also, for an arbitrary fuzzy topological space, a characterization is given of the set of all ultra fuzzy Compactifications.     

Finite groups in Stone-Cech compactifications

Let be an infinite cardinal and let G = ₂. Now let β Gbe the Stone–ech compactification of G as a discrete semigroup,and let =_<c_{β G} {xG\{0}:minsupp (x)}. We show that thesemigroup contains no nontrivial finite group.