Zero-dimensional Hausdorff Space Research Articles

Ideals of functions with compact support in the integer-valued case

Abstract For a zero-dimensional Hausdorff space X, denote, as usual, by C(X, ℤ) the ring of continuous integer-valued functions on X. If f ∈ C(X, ℤ), denote by Z(f) the set of all points of X that are mapped to 0 by f. The set $$\begin{array}{} \displaystyle C_K(X,\mathbb Z)=\{f\in C(X,\mathbb Z)\mid \text{cl}_X(X\smallsetminus \mathsf Z(f))\text{ is compact}\} \end{array}$$ is the integer-valued analogue of the ideal of functions with compact support in C(X). By first working in the category of locales and then interpreting the results in spaces, we characterize this ideal in several ways. Writing ζ X for the Banaschewski compactification of X, we also explore some properties of ideals of C(X, ℤ) associated with subspaces of ζ X analogously to how one associates, for any Tychonoff space Y, subsets of β Y with ideals of C(Y).

Two extensions of the stone duality to the category of zero-dimensional Hausdorff spaces

Extending the Stone Duality Theorem, we prove two duality theorems for the category ZHaus of zero-dimensional Hausdorff spaces and continuous maps. They extend also the Tarski Duality Theorem; the latter is even derived from one of them. We prove as well two new duality theorems for the category EDTych of extremally disconnected Tychonoff spaces and continuous maps. Also, we describe two categories which are dually equivalent to the category ZComp of zero-dimensional Hausdorff compactifications of zero-dimensional Hausdorff spaces and obtain as a corollary the Dwinger Theorem about zero-dimensional compactifications of a zero-dimensional Hausdorff space.

Some applications of the theory of Katětov order to ideal convergence

In this paper we establish connections between the theory of Katětov order on ideals on countable sets with ideal convergence in general topological spaces, which are used to study the following questions posed by X.G. Zhou, L. Liu and S. Lin [12]:(1)Whether every finite union of I-closed subsets is I-closed?(2)Must every I-continuous map preserve I-convergence?(3)Must every map preserving I-convergence preserve J-convergence if J⊆I? Our main results include:•If I is K-uniform, then every finite union of I-closed subsets is I-closed in any space;•there exists a countable zero-dimensional Hausdorff space with character equal to the continuum in which there are two I-closed sets with non-I-closed union for some tall Fσ-ideal I, while•it is independent of ZFC that for every Hausdorff space X of character less than the continuum and every tall Fσ-ideal I, finite unions of I-closed subsets are always I-closed in X.These answers Question (1). We show the answer to Question (2) is negative in ZFC and the answer to Question (3) is also negative if we replace J⊆I with J≤KI (which is weaker than J⊆I).

Tree-like constructions in topology and modal logic

Within ZFC, we develop a general technique to topologize trees that provides a uniform approach to topological completeness results in modal logic with respect to zero-dimensional Hausdorff spaces. Embeddings of these spaces into well-known extremally disconnected spaces then gives new completeness results for logics extending S4.2.

The Banaschewski compactification of an approach space is of Wallman-Shanin-type

For a topological zero-dimensional Hausdorff space (X, ) it is well known that the Banaschewski compactification ζ(X, ) is of Wallman-Shanin-type, meaning that there exists a closed basis (the collection of all clopen sets), such that the Wallman-Shanin compactification with respect to this closed basis is isomorphic to ζ(X, ). For an approach space (X, ) the Wallman-Shanin compactification W (X, ) with respect to a Wallman-Shanin basis (a particular basis of the lower regular function frame ) was introduced by R. Lowen and the second author. Recently, various constructions of the Banaschewski compactification known for a topological space were generalised to the approach case. Given a Hausdorff zero-dimensional approach space (X, ), constructions of the Banaschewski compactification ζ ∗(X, ) were developed by the authors. In this paper we construct a particular Wallman-Shanin basis for (X, ) and show that the Wallman-Shanin compactification with respect to this particular basis is isomorphic to ζ ∗(X, ).

Open and other kinds of map extensions over zero-dimensional local compactifications

Generalizing a theorem of Ph. Dwinger (1961) [7], we describe the partially ordered set of all (up to equivalence) zero-dimensional locally compact Hausdorff extensions of a zero-dimensional Hausdorff space. Using this description, we find the necessary and sufficient conditions which has to satisfy a map between two zero-dimensional Hausdorff spaces in order to have some kind of extension over arbitrary, but fixed, Hausdorff zero-dimensional local compactifications of these spaces; we consider the following kinds of extensions: continuous, open, quasi-open, skeletal, perfect, injective, surjective, dense embedding. In this way we generalize some classical results of B. Banaschewski (1955) [1] about the maximal zero-dimensional Hausdorff compactification. Extending a recent theorem of G. Bezhanishvili (2009) [2], we describe the local proximities corresponding to the zero-dimensional Hausdorff local compactifications.

Zero-dimensional proximities and zero-dimensional compactifications

We introduce zero-dimensional proximities and show that the poset 〈 Z ( X ) , ⩽ 〉 of inequivalent zero-dimensional compactifications of a zero-dimensional Hausdorff space X is isomorphic to the poset 〈 Π ( X ) , ⩽ 〉 of zero-dimensional proximities on X that induce the topology on X. This solves a problem posed by Leo Esakia. We also show that 〈 Π ( X ) , ⩽ 〉 is isomorphic to the poset 〈 B ( X ) , ⊆ 〉 of Boolean bases of X, and derive Dwinger's theorem that 〈 Z ( X ) , ⩽ 〉 is isomorphic to 〈 B ( X ) , ⊆ 〉 as a corollary. As another corollary, we obtain that for a regular extremally disconnected space X, the Stone–Čech compactification of X is a unique up to equivalence extremally disconnected compactification of X.

A generalized inductive limit strict topology on the space of bounded continuous functions

A generalized inductive limit strict topology β ∞ is defined on C b( X, E), the space of all bounded, continuous functions from a zero-dimensional Hausdorff space X into a locally K -convex space E, where K is a field with a nontrivial and nonarchimedean valuation, for which K is a complete ultrametric space. Many properties of the topology β ∞ are proved and the dual of ( C b ( X, E), β ∞) is studied.

The embeddability ordering of topological spaces

For K a set of topological spaces and X , Y ∈ K , the notation X ⊆ h Y means that X embeds homeomorphically into Y; and X ∼ Y means X ⊆ h Y ⊆ h X . With X ˜ : = { Y ∈ K : X ∼ Y } , the equivalence relation ∼ on K induces a partial order ⩽ h well-defined on K / ∼ as follows: X ˜ ⩽ h Y ˜ if X ⊆ h Y . For posets ( P , ⩽ P ) and ( Q , ⩽ Q ) , the notation ( P , ⩽ P ) ↪ ( Q , ⩽ Q ) means: there is an injection h : P → Q such that p 0 ⩽ P p 1 in P if and only if h ( p 0 ) ⩽ Q h ( p 1 ) in Q. For κ an infinite cardinal, a poset ( Q , ⩽ Q ) is a κ- universal poset if every poset ( P , ⩽ P ) with | P | ⩽ κ satisfies ( P , ⩽ P ) ↪ ( Q , ⩽ Q ) . The authors prove two theorems which improve and extend results from the extensive relevant literature. Theorem 2.2 There is a zero-dimensional Hausdorff space S with | S | = κ such that ( P ( S ) / ∼ , ⩽ h ) is a κ-universal poset. Theorem 3.1 There are a compact, connected Hausdorff space S and a set K of ( 2 κ -many) compact, connected subspaces of S such that (a) the posets ( P ( κ ) , ⊆ ) and ( K / ∼ , ⩽ h ) are isomorphic; and (b) ( K / ∼ , ⩽ h ) is a κ-universal poset. Further, one may arrange | S | = w ( S ) = | X | = w ( X ) = ℵ κ ⋅ c for each X ∈ K .

Hereditary Coreflective Subcategories of Categories of Topological Spaces

Let $\mathcal{A}$ be an epireflective subcategory of the category Top of topological spaces that is not bireflective (e.g., the category of Hausdorff spaces, the category of Tychonoff spaces) and ℬ be a coreflective subcategory of $\mathcal{A}$ . Extending the corresponding result obtained for coreflective subcategories of Top we prove that ℬ is hereditary if and only if it is closed under the formation of prime factors. As a consequence we obtain that every hereditary coreflective subcategory ℬ of $\mathcal{A}$ containing a non-discrete space is generated by a class of prime spaces and if $\mathcal{A}$ is a quotient-reflective subcategory of Top, then the assignment $\mathcal{B}\mapsto \mathcal{B}\cap \mathcal{A}$ gives a bijection of the collection of all hereditary coreflective subcategories of Top that contain the class FG of all finitely generated spaces onto the collection of all hereditary coreflective subcategories of $\mathcal{A}$ that contain $\mathbf{FG}\cap \mathcal{A}$ . Some applications of these results in the categories of Hausdorff spaces, Tychonoff spaces and zero-dimensional Hausdorff spaces are presented.

The maximal ring of quotientf-ring

In this article it is shown that the maximal quotient ringQA of a commutative semiprimef-ringA can be obtained by the formation of the orthocompletion ofA, followed by that of the classical quotient ring; for archimedeanf-rings the order of these can be inverted. It is shown that ifC=C(X, Z), whereX is a zero-dimensional Hausdorff space, then the integral closure is the Dedekind-McNeille completion ofC. The paper closes with a number of observations and examples.

A general theory of structure spaces with applications to spaces of prime ideals

A structure space is a quadrupleX=(X, d, A, P), where for some setR, X ⊂A=2 R ,d:X×X →A is defined byd(I, J)=J−I, andP is the family of cofinite subsets ofR. Forr e P, I e X, N r (I)={J e X: d(I, J) ⊂r},To(X)={Q ⊂X: if x e Q there is anr e P such thatN r (x) $$ \subseteq$$ Q}. ThenTo(X) is a (not usually Hausdorff) topology onX called the hull-kernel topology. Replacing d byd *, whered * (I, J)=d(J, I), or byd s, whered s (I, J.)=d(I, J) ∪d * (I, J), and proceeding in the obvious way yields thedual hull-kernel topology To(X *) andsymmetric topology To(X s ). The latter is always a zero-dimensional Hausdorff space. When R is a commutative ring with identity andX is a collection of proper prime ideals ofR, To(X s ) is usually called thepatch topology. Our generality enables us to improve on known results in the case of space of prime ideals and to apply this theory to a wide variety of algebraic structures. In particular, we establish criteria for a subspace of a structure space to be closed in the symmetric topology; we establish a duality between families of maximal elements in the hull-kernel topology and families of minimal elements in the dual hull-kernel topology of subspaces that are closed in the symmetric topology; we use topological constructions to generalize certain ring theoretic notions, such as radical ideals an annihilator ideals; we use this theory to obtain new results about subspaces of the space prime ideals of a reduced, commutative ring.

Orderability properties of a zero-dimensional space which is a continuous image of an ordered compactum

If X is a zero-dimensional Hausdorff space which is a continuous image of a compact linearly ordered topological space, then X can be embedded into some dendron. If, moreover, X is separable then X can be embedded into some arc. Both results can be treated as orderability theorems.

Bornological spaces of non-Archimedean valued functions with the compact-open topology

Let F be a field with nontrivial non-Archimedean valuation of rank one and let X be a zero-dimensional Hausdorff space. The vector space C ( X , F ) C(X,F) of all continuous functions from X into F is provided with the compact-open topology c. We prove that C ( X , F , c ) C(X,F,c) is bornological if and only if X is a Z-replete space.

A Boolean algebra without proper automorphisms

It is the purpose of this note to show that there exists an infinite Boolean algebra which has no proper automorphisms.' We shall construct a simply ordered set S, introduce a topology on this set in the usual manner, the so-called interval topology determined by the ordering relation,2 and prove that S is a compact zero-dimensional Hausdorff space and that the only homeomorphism on S onto S is the identity mapping. It is well known3 that the group of automorphisms of the set-field 3 consisting of all open and closed subsets of S is isomorphic to the group of all homeomorphisms on S onto S, whence it follows that B3 has no proper automorphisms. Consider a simply ordered set S with at least two elements. By the interval topology on S we mean the topology which has as a subbasis for open sets, the family of all sets UCS such that either