The partially ordered set of one-point extensions

A space Y is called an extension of a space X if Y contains X as a dense subspace. Two extensions of X are said to be equivalent if there is a homeomorphism between them which fixes X point-wise. For two (equivalence classes of) extensions Y and Y^′ of X let Y?Y^′ if there is a continuous function of Y^′ into Y which fixes X point-wise. An extension Y of X is called a one-point extension of X if Y?X is a singleton. Let P be a topological property. An extension Y of X is called a P-extension of X if it has P.One-point P-extensions comprise the subject matter of this article. Here P is subject to some mild requirements. We define an anti-order-isomorphism between the set of one-point Tychonoff extensions of a (Tychonoff) space X (partially ordered by ?) and the set of compact non-empty subsets of its outgrowth βX?X (partially ordered by ⊆). This enables us to study the order-structure of various sets of one-point extensions of the space X by relating them to the topologies of certain subspaces of its outgrowth. We conclude the article with the following conjecture. For a Tychonoff spaces X denote by U(X) the set of all zero-sets of βX which miss X.

Conjecture. For locally compact spaces X and Y the partially ordered sets(U(X),⊆)and(U(Y),⊆)are order-isomorphic if and only if the spacesclβX(βX?υX)andclβY(βY?υY)are homeomorphic.     

The enriched Vietoris monad on representable spaces

Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set monad becomes the filter monad, cocomplete ordered set translates to continuous lattice, distributivity means disconnectedness, and so on. Curiously, the dual(?) notion of completeness does not behave as the mirror image of the one of cocompleteness; and in this paper we have a closer look at complete spaces. In particular, we construct the “up-set monad” on representable spaces (in the sense of L. Nachbin for topological spaces, respectively C. Hermida for multicategories); we show that this monad is of Kock–Zöberlein type; we introduce and study a notion of weighted limit similar to the classical notion for enriched categories; and we describe the Kleisli category of our “up-set monad”. We emphasise that these generic categorical notions and results can be indeed connected to more “classical” topology: for topological spaces, the “up-set monad” becomes the lower Vietoris monad, and the statement “X   is totally cocomplete if and only if X^op$X^{\mathrm{op}}$ is totally complete” specialises to O. Wyler's characterisation of the algebras of the Vietoris monad on compact Hausdorff spaces as precisely the continuous lattices.     

Internal normality and internal compactness

This text contains an example which presents a way to modify any Dowker space to get a normal space X such that X×[0,1] is not κ-normal, and a theorem implying the existence of a non-Tychonoff space which is internally compact in a larger regular space. It gives answers to several questions by Arhangel'skii [A.V. Arhangel'skii, Relative normality and dense subspaces, Topology Appl. 123 (2002) 27-36].     

On order structure of the set of one-point Tychonoff extensions of a locally compact space

If a Tychonoff space X is dense in a Tychonoff space Y, then Y is called a Tychonoff extension of X. Two Tychonoff extensions Y₁ and Y₂ of X are said to be equivalent, if there exists a homeomorphism which keeps X pointwise fixed. This defines an equivalence relation on the class of all Tychonoff extensions of X. We identify those extensions of X which belong to the same equivalence classes. For two Tychonoff extensions Y₁ and Y₂ of X, we write Y₂?Y₁, if there exists a continuous function which keeps X pointwise fixed. This is a partial order on the set of all (equivalence classes of) Tychonoff extensions of X. If a Tychonoff extension Y of X is such that Y\X is a singleton, then Y is called a one-point extension of X. Let T(X) denote the set of all one-point extensions of X. Our purpose is to study the order structure of the partially ordered set (T(X),?). For a locally compact space X, we define an order-anti-isomorphism from T(X) onto the set of all nonempty closed subsets of βX\X. We consider various sets of one-point extensions, including the set of all one-point locally compact extensions of X, the set of all one-point Lindelöf extensions of X, the set of all one-point pseudocompact extensions of X, and the set of all one-point ?ech-complete extensions of X, among others. We study how these sets of one-point extensions are related, and investigate the relation between their order structure, and the topology of subspaces of βX\X. We find some lower bounds for cardinalities of some of these sets of one-point extensions, and in a concluding section, we show how some of our results may be applied to obtain relations between the order structure of certain subfamilies of ideals of C^∗(X), partially ordered with inclusion, and the topology of subspaces of βX\X. We leave some problems open.     

A pair of spaces of upper semi-continuous maps and continuous maps

For a Tychonoff space X, we use ↓USC(X) and ↓C(X) to denote the families of the regions below all upper semi-continuous maps and of the regions below all continuous maps from X to I=[0,1], respectively. In this paper, we consider the spaces ↓USC(X) and ↓C(X) topologized as subspaces of the hyperspace Cld(X×I) consisting of all non-empty closed sets in X×I endowed with the Vietoris topology. We shall prove that ↓USC(X) is homeomorphic (≈) to the Hilbert cube Q=ω[−1,1] if and only if X is an infinite compact metric space. And we shall prove that (↓USC(X),↓C(X))≈(Q,c₀), where , if and only if ↓C(X)≈c₀ if and only if X is a compact metric space and the set of isolated points is not dense in X.     

Some upper bounds for density of function spaces

Let Cα(X,Y) be the set of all continuous functions from X to Y endowed with the set-open topology where α is a hereditarily closed, compact network on X which is closed under finite unions. We proved that the density of the space Cα(X,Y) is at most iw(X)⋅d(Y) where iw(X) denotes the i-weight of the Tychonoff space X, and d(Y) denotes the density of the space Y when Y is an equiconnected space with equiconnecting function Ψ, and Y has a base consists of Ψ-convex subsets of Y. We also prove that the equiconnectedness of the space Y cannot be replaced with pathwise connectedness of Y. In fact, it is shown that for each infinite cardinal κ, there is a pathwise connected space Y such that π-weight of Y is κ, but Souslin number of the space Ck([0,1],Y) is κ².     

Compactness in the fine topology

For reasonable spaces (including topological manifolds) X, Y, we characterize compact subsets of the space of continuous maps from X to Y, topologized with the fine (Whitney) C⁰-topology. In the case of smooth manifolds, we characterize also compact subsets of the space of C^r maps in the Whitney C^r topology.     

Fodor-type Reflection Principle and reflection of metrizability and meta-Lindelöfness

We introduce a new reflection principle which we call “Fodor-type Reflection Principle” (FRP). This principle follows from but is strictly weaker than Fleissner's Axiom R. For instance, FRP does not impose any restriction on the size of the continuum, while Axiom R implies that the continuum has size ?ℵ₂.We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelöf if all of its subspaces of cardinality ?ℵ₁ are (Theorem 4.3). It follows that, under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality ?ℵ₁ are (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R.We also give several other results in this vein, some in ZFC, others in some further extension of ZFC. For example, we prove in ZFC that if X is a locally (countably) compact space of singular cardinality in which every subspace of smaller size is metrizable then X itself is also metrizable (Corollary 5.2).     

D-spaces and thick covers

The following results are obtained.

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An open neighbornet U of X has a closed discrete kernel if X has an almost thick cover by countably U-close sets.

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Every hereditarily thickly covered space is aD and linearly D.

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Every t-metrizable space is a D-space.

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X is a D-space if X has a cover {Xα:α<λ} by D-subspaces such that, for each β<λ, the set ?{Xα:α<β} is closed.

     

Lelek maps and n-dimensional maps from compacta to polyhedra

For a natural number m?0, a map from a compactum X to a metric space Y is an m-dimensional Lelek map if the union of all non-trivial continua contained in the fibers of f is of dimension ?m. In [M. Levin, Certain finite-dimensional maps and their application to hyperspaces, Israel J. Math. 105 (1998) 257-262], Levin proved that in the space C(X,I) of all maps of an n-dimensional compactum X to the unit interval I=[0,1], almost all maps are (n−1)-dimensional Lelek maps. Moreover, he showed that in the space C(X,Ik) of all maps of an n-dimensional compactum X to the k-dimensional cube Ik (k?1), almost all maps are (n−k)-dimensional Lelek maps. In this paper, we generalize Levin's result. For any (separable) metric space Y, we define the piecewise embedding dimension ped(Y) of Y and we prove that in the space C(X,Y) of all maps of an n-dimensional compactum X to a complete metric ANR Y, almost all maps are (n−k)-dimensional Lelek maps, where k=ped(Y). As a corollary, we prove that in the space C(X,Y) of all maps of an n-dimensional compactum X to a Peano curve Y, almost all maps are (n−1)-dimensional Lelek maps and in the space C(X,M) of all maps of an n-dimensional compactum X to a k-dimensional Menger manifold M, almost all maps are (n−k)-dimensional Lelek maps. It is known that k-dimensional Lelek maps are k-dimensional maps for k?0.     

More stratifiable function spaces

If X is a compact-covering image of a closed subspace of product of a σ-compact Polish space and a compact space, then Ck(X,M), the space of continuous maps of X into M with the compact-open topology, is stratifiable for any metric space M.If X is σ-compact Polish, K is compact and M metric then every point of Ck(X×K,M) has a closure-preserving local base, and hence this function space is M₁.     

The connected Vietoris powerlocale

The connected Vietoris powerlocale is defined as a strong monad Vc on the category of locales. VcX is a sublocale of Johnstone's Vietoris powerlocale VX, a localic analogue of the Vietoris hyperspace, and its points correspond to the weakly semifitted sublocales of X that are “strongly connected”. A product map ×:VcX×VcY→Vc(X×Y) shows that the product of two strongly connected sublocales is strongly connected. If X is locally connected then VcX is overt. For the localic completion of a generalized metric space Y, the points of are certain Cauchy filters of formal balls for the finite power set FY with respect to a Vietoris metric.Application to the point-free real line R gives a choice-free constructive version of the Intermediate Value Theorem and Rolle's Theorem.The work is topos-valid (assuming natural numbers object). Vc is a geometric construction.     

A note on a theorem of Talagrand

We provide an elementary argument to show that if for a hemicompact kR-space X the space Cp(X) contains a subset S which separates the points of X and is dominated by irrationals, i.e. S is covered by a family of compact sets such that Kα⊂Kβ for α?β, then Cp(X) is also dominated by irrationals; consequently Cp(X) is K-analytic. This fact (which fails for non-hemicompact spaces X) extends an old result of Talagrand.     

Transfinite Hausdorff dimension

Making extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorphisms), in fact like Hausdorff dimension, it does not increase under Lipschitz maps, and it also satisfies the intermediate dimension property (Theorem 2.7). The primary goal of transfinite Hausdorff dimension is to classify metric spaces with infinite Hausdorff dimension. Indeed, if tHD(X)?ω₀, then HD(X)=+∞. We prove that tHD(X)?ω₁ for every separable metric space X, and, as our main theorem, we show that for every ordinal number α<ω₁ there exists a compact metric space Xα (a subspace of the Hilbert space l₂) with tHD(Xα)=α and which is a topological Cantor set, thus of topological dimension 0. In our proof we develop a metric version of Smirnov topological spaces and we establish several properties of transfinite Hausdorff dimension, including its relations with classical Hausdorff dimension.     

Certain sequences with compact closure

This paper deals with a question which is stated by quite simple definitions. A sequence {xn} in a space X is called a β-sequence if every subsequence of it has a cluster point in X. The closure of the sequence {xn} means the closure of in X. Here we consider the question when a β-sequence has compact closure. We give several answers to this question.     

Rectangular products with ordinal factors

Let A and B be subspaces of an ordinal. It is proved that the product A×B is countably paracompact if and only if it is rectangular. Before this main result, we discuss several covering properties of products with one ordinal factor. In particular, for every paracompact space X, it is proved that the product X×A is paracompact if so is A.     

On the tychonoff functor and w-compactness

The main purpose of this paper is to settle the following problem concerning a product formula for the Tychonoff functor τ, by introducing the notion of w-compact spaces: Characterize a topological space X such that τ(X×Y)=τ(X)×τ(Y) for any topological space Y. We also study the properties of w-compact spaces, and it is proved that, for any family {X_α} of w-compact spaces, the product ΠX_α is also w-compact and τ(ΠX_α)=Πτ(X_α).     

Convexity of momentum maps: A topological analysis

The Local-to-Global-Principle used in the proof of convexity theorems for momentum maps has been extracted as a statement of pure topology enriched with a structure of convexity. We extend this principle to not necessarily closed maps f:X→Y where the convexity structure of the target space Y need not be based on a metric. Using a new factorization of f, convexity of the image is proved without local fiber connectedness, and for arbitrary connected spaces X.