In this paper we generalize the notion of perfect compactification of a Tychonoff space to a generic extension of any space by introducing the concept of perfect pair. This allow us to simplify the treatment in a basic way and in a more general setting. Some [S], [S], and [D]’s results are improved and new characterizations for perfect (Hausdorff) extensions of spaces are obtained.
The paper is devoted to the study of the ordered set of all, up to equivalence, -compactifications of an Alexandroff space . The notion of -weight (denoted by ) of an Alexandroff space is introduced and investigated. Using results in ([7]) and ([5]), lattice properties of and are studied, where is the set of all, up to equivalence, -compactifications of for which . A characterization of the families of bounded functions generating an -compactification of is obtained. The notion...
The completion of a Suslin tree is shown to be a consistent example of a Corson compact L-space when endowed with the coarse wedge topology. The example has the further properties of being zero-dimensional and monotonically normal.
In the theory of compactifications, Magill's theorem that the continuous image of a remainder of a space is again a remainder is one of the most important theorems in the field. It is somewhat unfortunate that the theorem holds only in locally compact spaces. In fact, if all continuous images of a remainder are again remainders, then the space must be locally compact. This paper is a modification of Magill's result to more general spaces. This of course requires restrictions on the nature of the...
For X a Tikhonov space, let F(X) be the algebra of all real-valued continuous functions on X that assume only finitely many values outside some compact subset. We show that F(X) generates a compactification γX of X if and only if X has a base of open sets whose boundaries have compact neighborhoods, and we note that if this happens then γX is the Freudenthal compactification of X. For X Hausdorff and locally compact, we establish an isomorphism between the lattice of all subalgebras of and the...
Let X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)ℕ of the Hilbert cube Q = [−1, 1]ℕ. In this paper, generalizing the Sakai-Uehara’s result to the non-compact case, we construct a natural compactification (X) of C(X) such that the pair ( (X), C(X)) is homeomorphic to (Q, s). In case...
A new form of -compactness is introduced in -topological spaces by -open -sets and their inequality where is a complete de Morgan algebra. It doesn’t rely on the structure of the basis lattice . It can also be characterized by means of -closed -sets and their inequality. When is a completely distributive de Morgan algebra, its many characterizations are presented and the relations between it and the other types of compactness are discussed. Countable -compactness and the -Lindelöf property...