Abstract:
We study the behavior of certain spaces and their compactificability classes at infinity. Among other results we show that every noncompact, locally compact, second countable Hausdorff space X such that each neighborhood of infinity (in the Alexandroff compactification) is uncountable, has 𝒞(X)=𝒞(ℝ). We also prove some criteria for (non-) comparability of the studied classes of mutual compactificability.

Abstract:
We apply the theory of the mutual compactificability to some spaces, mostly derived from the real line. For example, any noncompact locally connected metrizable generalized continuum, the Tichonov cube without its zero point Iℵ0\{0}, as well as the Cantor discontinuum without its zero point Dℵ0\{0} are of the same class of mutual compactificability as ℝ.

Abstract:
Two disjoint topological spaces X, Y are (T2-) mutually compactificable if there exists a compact (T2-) topology on K=X∪Y which coincides on X, Y with their original topologies such that the points x∈X, y∈Y have open disjoint neighborhoods in K. This paper, the first one from a series, contains some initial investigations of the notion. Some key properties are the following: a topological space is mutually compactificable with some space if and only if it is θ-regular. A regular space on which every real-valued continuous function is constant is mutually compactificable with no S2-space. On the other hand, there exists a regular non-T3.5 space which is mutually compactificable with the infinite countable discrete space.

Abstract:
Two disjoint topological spaces X, Y are mutually compactificable if there exists a compact topology on K=X∪Y which coincides on X, Y with their original topologies such that the points x∈X, y∈Y have disjoint neighborhoods in K. The main problem under consideration is the following: which spaces X, Y are so compatible such that they together can form the compact space K? In this paper we define and study the classes of spaces with the similar behavior with respect to the mutual compactificability. Two spaces X1, X2 belong to the same class if they can substitute each other in the above construction with any space Y. In this way we transform the main problem to the study of relations between the compactificability classes. Some conspicuous classes of topological spaces are discovered as the classes of mutual compactificability. The studied classes form a certain “scale of noncompactness” for topological spaces. Every class of mutual compactificability contains a T1 representative, but there are classes with no Hausdorff representatives.

Abstract:
We show that there exists a canonical topology, naturally connected with the causal site of J. D. Christensen and L. Crane, a pointless algebraic structure motivated by quantum gravity. Taking a causal site compatible with Minkowski space, on every compact subset our topology became a reconstruction of the original topology of the spacetime (only from its causal structure). From the global point of view, the reconstructed topology is the de Groot dual or co-compact with respect to the original, Euclidean topology. The result indicates that the causality is the primary structure of the spacetime, carrying also its topological information.

Abstract:
The bitopological unstability of RR-pairwise paracompactness inpresence of pairwise Hausdorff separation axiom is caused by abitopological property which is much weaker and more local thanRR-pairwise paracompactness. We slightly generalize someMichael's constructions and characterize RR-pairwiseparacompactness in terms of bitopological θ-regularity, andsome other weaker modifications of pairwise paracompactness.

Abstract:
A considerable problem of some bitopological covering properties is the bitopological unstability with respect to the presence of the pairwise Hausdorff separation axiom. For instance, if the space is RR-pairwise paracompact, its two topologies will collapse and revert to the unitopological case. We introduce a new bitopological separation axiom τS2σ which is appropriate for the study of the bitopological collapse. We also show that the property that may cause the collapse is much weaker than some modifications of pairwise paracompactness and we generalize several results of T. G. Raghavan and I. L. Reilly (1977) regarding the comparison of topologies.

Abstract:
We modify the concept of θ-regularity forspaces with 2 and 3 topologies. The new, more general property isfully preserved by sums and products. Using some bitopologicalreductions of this property, Michael's theorem for several variantsof bitopological paracompactness is proved.

Abstract:
In this paper we give an embedding characterization of -regularity using the Wallman-type compactlfication. The productivity of -regularity and a slight generalization of Nagami's Product Theorem to non-Hausdorff paracompact ￠ ‘-spaces we obtain as a corollary.

Abstract:
In this paper we study -regularity and its relations to other topological properties. We show that the concepts of -regularity (Jankovi , 1985) and point paracompactness (Boyte, 1973) coincide. Regular, strongly locally compact or paracompact spaces are -regular. We discuss the problem when a (countably) -regular space is regular, strongly locally compact, compact, or paracompact. We also study some basic properties of subspaces of a -regular space. Some applications: A space is paracompact iff the space is countably -regular and semiparacompact. A generalized F -subspace of a paracompact space is paracompact iff the subspace is countably -regular.