Wallman-type Compactification Research Articles

Normal base of ditopological texture space and its Wallman-type compactification

In this study, it has been intended to create an approach providing to get Wallman-type compacti cation of ditopological texture space. For this purpose, the concept of normal basewell-known in general topology is relocated to ditopological texture space. We get the Wallmancompacti cation of ditopological texture space by de ning normal base ,co-normal base.

Fan-Gottesman compactification of some specific spaces is Wallman-type compactification

It is known that some compactification methods have similar properties when studied spaces are local compact hausdorff. But this compactification methods (one-point compactification, Stone-Cech compactification, Wallman compactification, Fan-Gottesman compactification, Freudenthal compactification, etc.) have very few similarity in some specific spaces. In this paper especially it is investigated relation between Fan-Gottesman and Wallman compactifications and it is shown that Fan-Gottesman compactification of some interesting and specific spaces such as normal A 2 and T 4 is Wallman-type compactification.

A quasi-uniform characterization of Wallman type compactifications

A *-compactification of aT1quasi-uniform space (X,U) is a compactT1quasi-uniform space (Y,V) that has aT(V*)-dense subspace quasi-isomorphic to (X,U), whereV* denotes the coarsest uniformity finer thanV.In this paper we characterize all Wallman type compactifications of aT1topological space in terms of the *-compactification of its point symmetric totally bounded transitive compatible quasi-uniformities. We deduce that the *-compactification of the Pervin quasi-uniformity of any normalT1topological spaceXis exactly the Stone-Cech compactification ofX. We also obtain a characterization of those Hausdorff compactifications of a given space, which are of Wallman type.

Hausdorff compactifications and zero-one measures II

<p>The notion of PBS-sublattice is introduced and, using it, a simplification of the results of [6] and of some results of [5] is obtained. Two propositions concerning Wallman-type compactifications are presented as well.</p>

Hausdorff compactifications and zero-one measures

It is well known that the Wallman-type compactifications of a Tychonoff space X can be obtained as spaces of all regular zero-one measures on suitable lattices of subsets of X (see [1, 2, 4, 12]). Using the technique developed in [5, 6], we find for any Tychonoff space X a Boolean algebra [Bscr ]X and a set [Lscr ]X of sublattices of [Bscr ]X having the following property: for any Hausdorff compactification cX of X there exists a (unique) LcX ∈ [Lscr ]X such that the maximal spectrum of LcX and the space of all u-regular zero-one measures on the Boolean subalgebra b(LcX) of [Bscr ]X, generated by LcX, are Hausdorff compactifications of X equivalent to cX. Let us give more details now.

A Wallman-Shanin-Type Compactification for Approach Spaces

In [11] a Cech-Stone-type compactification theory was developed for UAP2. In this paper we construct a Wallman-Shanin-type compactification theory for weakly symmetric T1 approach spaces which form a full subcategory of AP properly containing UAP2. For a weakly symmetric T1 approach space, we also investigate the relation between the topological bicoreflection of its Wallman-type compactification and the classical Wallman compactification of its topological bicoreflection, and we show that our theory extends the classical topological Wallman compactification theory. It is shown in [14] that our present theory also extends the CechStone-type theory from [11].

Compactifications and quotient lattices of Wallman bases

We improve the intuition and some ideas of G. D. Faulkner and M. C. Vipera presented in the article “Remainders of compactifications and their relation to a quotient lattice of the topology” [Proc. Amer. Math. Soc. 122 (1994)] in connection with the question about internal conditions for locally compact spaces X and Y under which βX/X ≅ βY/Y or, more generally, under which the remainders of compactifications of X belong to the collection of the remainders of compactifications of Y. We point out the reason why the quotient lattice of the topology considered by Faulkner and Vipera cannot lead to a satisfactory answer to the above question. We replace their lattice by the qoutient lattice of a new equivalence relation on a Wallman base in order to describe a method of constructing a Wallman-type compactification which allows us to deduce more complete solutions to the problems investigated by Faulkner and Vipera.

L-fuzzy Wallman-type compactification

In this paper, the concepts of α-ultra-closed filter and α-finite intersection property are introduced into L-fuzzy topological space. By using these two concepts, the Nice-compactness is characterized again. Then, we construct a fuzzy T 1-compactification of a T 1- L-fuzzy topological space, and the crisp Wallman-type compactification is extended to L-fuzzy topological space for the first time.

A bitopological view on cocompact extensions

Let (X, V, W) be a bitopological space. The topology V is a (fine) cotopology of W if V ⊂ W, V is a T1-topology and W is (completely) regular with respect to V.Let (X, V, W) be a bitopological space such that V is a fine cotopology of W. Related to special closed subbases S and T of V and W respectively, a construction of a pairwise dense embedding e: (X, V, W) → (X, V, W) is presented such that V is a compact fine cotopology of W.The main properties of our construction are the following.1.(1) If the topology W is compact, then V = W and (X, W) is a Wallman-type compactification of (X, W). Every Wallman-type compactification can be reproduced in this way.2.(2) If the topology W̄ is metrizable, then (X, W) is a topologically complete extension of (X, W). Every topologically complete metrizable extension can be reproduced by this construction.

Lattice measures and topologies

We abstract Frink's notion of a normal base of a topological space to an arbitrary lattice, and replace the notion of filters on a base by zero-one measures on a lattice. This offers analytical simplification and clarijication, and extends to arbitrary measures as well. By putting a topology on the set of measures, we generalize the notion of Wallman-type compactifications, and we look at relations between the compactifications by examining the underlying lattices.

Round subsets of Wallman-type compactifications

Let ℒ be a normal base of a Tychonoff spaceXand ωℒ denote the Wallman-type (real-) compactification ofXgenerated by ℒ. This Wallman-type compactification is known to associate with a unique proximity δ. A ℒ-filter ℒ is round if for eachF∈ ℒ there is anFo∈ ℒ there is anFo∈ ℒ such thatFo(X-F). A subsetAof ω(£) is called a round subset of ω (£) iff for eachZ∈ ℒ, ifC1w(x)Zcontains A, then it is a neighborhood ofA. Properties of round ℒ-filters and round sets of ω(ℒ) are introduced. We also prove that the intersection of all the free ℒ-ultrafilters is ℒ= {Z∈ ℒ: C1x(X-Z) is compact} iff ω(ℒ) –Xis a round subset of ω(ℒ) if ℒ is a separating nest generated intersection ring with property (α) then ω(ℒ) -v(ℒ) is a round subset of ω(ℒ).

Nonstandard topological extensions

This paper investigates the nonstandard theory of filters on a non-empty meet-semi-lattice of sets and applies this theory to the general study of topological extensionsYfor a spaceX.In particular, we apply this theory to Baire and quasi-H-closed extensions as well as Wallman type compactifications. Whereas these extensions have previously teen obtained and studied as types of ultrafilter extensions, we study them as subsets of an enlargement ofX.SinceX⊂Y⊂ ◯ and the elements ofXandY-Xare of the same set-theoretic type, these extensions appear more natural from the nonstandard viewpoint.

Wallman t 1 -compactifications as epireflections

The purpose of this paper is to generalize the recent results of Harris concerning the functorial properties of the Wallman compactification of a T 1 -space. Steiner has introduced the Wallman-type compactifications of T 1 -spaces endowed with separating bases. We introduce a definition of morphism between such spaces to obtain a category which we denote by wosep. We prove that the Wallman-type compactification on objects can be extended to a functor on wosep and that the compact objects give an epireflective subcategory of wosep.

Wallman-type compactifications on 0-dimensional spaces

Let E E be Hausdorff 0 0 -dimensional, D \mathcal {D} the discrete space { 0 , 1 } \{0, 1\} , and N \mathcal {N} the discrete space of all nonnegative integers. Every E E -completely regular space X X has a clopen normal base F \mathcal {F} with X ∖ F ∈ F X\backslash F \in \mathcal {F} for each F ∈ F F \in \mathcal {F} . The Wallman compactification ω ( F ) \omega (\mathcal {F}) is D \mathcal {D} -compact. Moreover, if an E E -completely regular space X X has a countably productive clopen normal base F \mathcal {F} with X ∖ F ∈ F X\backslash F \in \mathcal {F} for each F ∈ F F \in \mathcal {F} , then the Wallman space η ( F ) \eta (\mathcal {F}) is N \mathcal {N} -compact. Hence, if X X has such an F \mathcal {F} , and is an F \mathcal {F} -realcompact space, then X X is N \mathcal {N} -compact.

A note on 𝒵-realcompactifications

Orrin Frink showed that the real-valued functions over a Tychonoff space X which may be continuously extended to ω ( Z ) \omega (\mathcal {Z}) , the Wallman-type compactification associated with a normal base Z \mathcal {Z} for X, are those which are Z \mathcal {Z} -uniformly continuous Let Z \mathcal {Z} be a delta normal base on a Tychonoff space X, and let η ( Z ) \eta (\mathcal {Z}) be the corresponding Z \mathcal {Z} -realcompactification of X. In this note we show that countable Z \mathcal {Z} -uniform continuity is a sufficient but not a necessary condition for continuously extending real-valued functions over X to η ( Z ) \eta (\mathcal {Z}) .

Wallman-Type Compactifications and Products

Y is a Wallman-type compactification (O. Frink, Amer. J. Math. 86 (1964), 602-607) of X in case there is a normal base Z for the closed sets of X such that the ultrafilter space from Z, denoted $\omega (Z)$, is topologically Y. It is not known if every compactification is Wallman-type. For ${Z_\alpha }$ a normal base for the closed sets of ${X_\alpha }$ for each a belonging to an index set $\Delta$ it is shown that the Tychonoff product space ${\prod _{\alpha \in \Delta }}\omega ({Z_\alpha })$ is a Wallman compactification of ${\prod _{\alpha \in \Delta }}{X_\alpha }$. Also for $X \subset T \subset \omega (Z)$ with Z a normal base for the closed sets of X, a proof that $\omega (Z)$ is a Wallman-type compactification of T is indicated.

Wallman-type compactifications and products

Y is a Wallman-type compactification (O. Frink, Amer. J. Math. 86 (1964), 602-607) of X in case there is a normal base Z for the closed sets of X such that the ultrafilter space from Z, denoted ω ( Z ) \omega (Z) , is topologically Y. It is not known if every compactification is Wallman-type. For Z α {Z_\alpha } a normal base for the closed sets of X α {X_\alpha } for each a belonging to an index set Δ \Delta it is shown that the Tychonoff product space ∏ α ∈ Δ ω ( Z α ) {\prod _{\alpha \in \Delta }}\omega ({Z_\alpha }) is a Wallman compactification of ∏ α ∈ Δ X α {\prod _{\alpha \in \Delta }}{X_\alpha } . Also for X ⊂ T ⊂ ω ( Z ) X \subset T \subset \omega (Z) with Z a normal base for the closed sets of X, a proof that ω ( Z ) \omega (Z) is a Wallman-type compactification of T is indicated.

Gelfand and Wallman-type compactifications

Gelfand and Wallman-type compactifications