Homogeneous Hausdorff Research Articles

On Wilder, Strongly Wilder, Continuumwise Wilder, D, D∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^*$$\\end{document}, and D∗∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^{**}$$\\end{document} Continua

We study properties of Wilder, strongly Wilder, continuumwise Wilder, D, D∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^*$$\\end{document}, and D∗∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^{**}$$\\end{document} Hausdorff continua. We present an example of a colocally connected continuum that is not a D∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^*$$\\end{document}-continuum, answering a question by Espinoza and Matsuhashi. We give several positive answers to this question for unicoherent continua. We also present some equivalences for the class of homogeneous Hausdorff continua with the property of Kelley.

Power homogeneous compacta and variations on tightness

The weak tightness wt(X), introduced in [6], has the property wt(X)≤t(X). It was shown in [4] that if X is a homogeneous compactum then |X|≤2wt(X)πχ(X). We introduce the almost tightness at(X) with the property wt(X)≤at(X)≤t(X) and show that if X is a power homogeneous compactum then |X|≤2at(X)πχ(X). This improves the result of Arhangel′skiĭ, van Mill, and Ridderbos in [2] that |X|≤2t(X) for a power homogeneous compactum X and gives a partial answer to a question in [4]. In addition, if X is a homogeneous Hausdorff space we show that |X|≤2pwcL(X)wt(X)πχ(X)pct(X), improving a result in [3]. It also extends the result in [4] into the Hausdorff setting. The cardinal invariant pwLc(X), introduced in [5] by Bella and Spadaro, satisfies pwLc(X)≤L(X) and pwLc(X)≤c(X). We also show the weight w(X) of a homogeneous space X is bounded in various contexts using wt(X). One such result is that if X is homogeneous and regular then w(X)≤2L(X)wt(X)pct(X). This generalizes a result in [4] that if X is a homogeneous compactum then w(X)≤2wt(X).

Cardinality bounds via covers by compact sets

We establish results concerning covers of spaces by compact and related sets. Several cardinality bounds follow as corollaries. Introducing the cardinal invariant $$\overline{\psi}_c(X)$$ , we show that $$|X|\leq \pi\chi(X)^{c(X)\overline{\psi}_c(X)}$$ for any topological space X. If X is Hausdorff then $$\overline{\psi}_c(X)\leq\psi_c(X)$$ ; this gives a strengthening of a theorem of Shu-Hao [24]. We also prove that $$|X|\leq 2^{pwL_c(X)t(X)pct(X)}$$ for a homogeneous Hausdorff space X. The invariant $$pwL_c(X)$$ , introduced in [9], is bounded above by both L(X) and c(X). Our result thus improves the bound $$|X|\leq 2^{L(X)t(X)pct(X)}$$ for homogeneous Hausdorff spaces X [13] and represents a new extension of de la Vega's Theorem [15] into the Hausdorff setting. Moreover, we show $$pwL(X)\leq aL(X)$$ , demonstrating that $$2^{pwL(X)\chi(X)}$$ is not a cardinality bound for all Hausdorff spaces. This answers a question of Bella and Spadaro [9]. A further theorem on covers by $$G^c_\kappa$$ -sets lead to cardinality bounds involving the linear Lindelof degree $$lL(X)$$ , a weakening of L(X). It was shown in [5] that $$|X|\leq 2^{lL(X)F(X)\psi(X)}$$ for Tychonoff spaces. We show the consistency of a) $$|X|\leq 2^{lL(X)F(X)\psi_c(X)}$$ if X is Hausdorff, and b) $$|X|\leq 2^{lL(X)F(X)pct(X)}$$ if X is Hausdorff and homogeneous. If X is additionally regular, the former consistently improves the result from [5]. The latter gives a consistent improvement of the inequality $$|X|\leq 2^{L(X)t(X)pct(X)}$$ for homogeneous Hausdorff spaces.

There are arbitrarily large uniquely homogeneous spaces

A topological space X is uniquely homogeneous if for every a,b∈X, there is exactly one homeomorphism h:X→X such that h(a)=b. We say that X is strongly uniquely homogeneous if it is uniquely homogeneous and the only continuous functions f:X→X are homeomorphisms and constant functions. We show that if n≥2, then there is a strongly uniquely homogeneous subspace of Rn, all of whose homeomorphisms are restrictions of isometries of Rn (rigid motions if n is even). Using a similar construction, we show that if κ is an infinite cardinal, then there is a strongly uniquely homogeneous Hausdorff and completely regular space of cardinality 2κ.

On cardinality bounds involving the weak Lindelöf degree

We give a general closing-off argument in Theorem 2.3 from which several corollaries follow, including (1) if X is a locally compact Hausdorff space then |X| ≤ 2wL(X)ψ(X), and (2) if X is a locally compact power homogeneous Hausdorff space then |X| ≤ 2wL(X)t(X). The first extends the well-known cardinality bound 2ψ(X) for a compactum X in a new direction. As |X| ≤ 2wL(X)χ(X) for a normal spaceX[4], this enlarges the class of known Tychonoff spaces for which this bound holds. In 2.12 we give a short, direct proof of (1) that does not use 2.3. Yet 2.3 is broad enough to establish results much more general than (1), such as if X is a regular space with a π-base ℬ such that |B| ≤ 2wL(X)χ(X) for all B ∈ ℬ, then |X| ≤ 2wL(X)χ(X).Separately, it is shown that if X is a regular space with a π-base whose elements have compact closure, then |X| ≤ 2wL(X)ψ(X)t(X). This partially answers a question from [4] and gives a third, separate proof of (1). We also show that if X is a weakly Lindelöf, normal, sequential space with χ(X) ≤ 2ℵ0, then |X| ≤ 2ℵ0.Result (2) above is a new generalization of the cardinality bound 2t(X) for a power homogeneous compactum X (Arhangel'skii, van Mill, and Ridderbos [3], De la Vega in the homogeneous case [10]). To this end we show that if U ⊆ clD ⊆ X, where X is power homogeneous and U is open, then |U| ≤ |D|πχ(X). This is a strengthening of a result of Ridderbos [19].

The weak Lindelöf degree and homogeneity

In this note a unifying closing-off argument is given (Theorem 2.5) involving the weak Lindelöf degreewL(X) of a Hausdorff space X and covers of X by compact subsets. This has among it corollaries the known cardinality bound 2wLc(X)χ(X) for spaces with Urysohn-like properties (Alas, 1993, [1], Bonanzinga, Cammaroto, and Matveev, 2011, [9]), the known bound cardinality bound 2wL(X)χ(X) for spaces with a dense set of isolated points (Dow and Porter, 1982, [14]), and two new cardinality bounds for power homogeneous spaces. In particular, it is shown that (a) if X is a power homogeneous Hausdorff space that is either quasiregular or Urysohn, then |X|⩽2wLc(X)t(X)pct(X), and (b) if X is a power homogeneous Hausdorff space with a dense set of isolated points then |X|⩽2wL(X)t(X)pct(X). These two bounds represent improvements on bounds for power homogeneous spaces given in Carlson et al. (2012) [11], as wLc(X)⩽aLc(X) for any space X. These results establish that known cardinality bounds for spaces with Urysohn-like properties, as well as spaces with a dense set of isolated points, are consequences of more general results that also give “companion” bounds for power homogeneous spaces with these properties.

On cardinality bounds for homogeneous spaces and the [formula omitted]-modification of a space

Improving a result in Carlson and Ridderbos (2012) [9], we construct a closing-off argument showing that the Lindelöf degree of the Gκ-modification of a space X is at most 2L(X)F(X)⋅κ, where F(X) is the supremum of the lengths of all free sequences in X and κ is an infinite cardinal. From this general result follow two corollaries: (1) |X|⩽2L(X)F(X)pct(X) for any power homogeneous Hausdorff space X, where pct(X) is the point-wise compactness type of X, and (2) |X|⩽2L(X)F(X)ψ(X) for any Hausdorff space X, as shown recently by Juhász and Spadaro (preprint) [17]. By considering the Lindelöf degree of the related Gκc-modification of a space X, we also obtain two consequences: (1) if X is a power homogeneous Hausdorff space then |X|⩽2aLc(X)t(X)pct(X), where aLc(X) is the almost Lindelöf degree with respect to closed sets, and (2) |X|⩽2aLc(X)t(X)ψc(X) for any Hausdorff space X, a well-known result of Bella and Cammaroto (1988) [4]. This demonstrates that both the Juhász–Spadaro and Bella–Cammaroto cardinality bounds for Hausdorff spaces are consequences of more general results that additionally lead to companion bounds for power homogeneous Hausdorff spaces. Finally, we give cardinality bounds for θ-homogeneous spaces that generalize those for homogeneous spaces, including cases in which the Hausdorff condition is relaxed.

On the cardinality of power homogeneous Hausdorff spaces

We prove that the cardinality of power homogeneous Hausdorff spaces X is bounded by d(X) ��(X) . This inequality improves many known results and it also solves a question by J. van Mill. We further introduce �-power homogeneity, which leads to a new proof of van Douwen's theorem.

A new bound on the cardinality of homogeneous compacta

We show (in ZFC) that if X is a compact homogeneous Hausdorff space then | X | ⩽ 2 t ( X ) , where t ( X ) denotes the tightness of X. It follows that under GCH the character and the tightness of such a space coincide.

Continuously homogeneous continua and their arc components

Let X X be a continuously homogeneous Hausdorff continuum. We prove that if there is a sequence A 1 , A 2 , … {A_1},{A_2}, \ldots of its arc components with X = c1 A 1 ∪ c1 A 2 ∪ ⋯ X = {\text {c1}}{A_1} \cup {\text {c1}}{A_2} \cup \cdots , and there is an arc component of X X with nonempty interior, then X X is arcwise connected. As consequences and applications we get: (1) if X X is the countable union of arcwise connected continua, then X X is arcwise connected; (2) if X X is nondegenerate and metric, the number of its arc components is countable and it contains no simple triod, then it is either an arc or a simple closed curve; and, in particular, (3) an arc is the only nondegenerate continuously homogeneous arc-like metric continuum with countably many arc components.

The sequential defect of the cross topology is ω 1

In response to a question by A. Arhangel'skiǐ and S.P. Franklin, we give an example of a countable homogeneous Hausdorff sequential space in which the topological closure is obtained by precisely ω 1 iterations of the sequential closure.

A short proof of a cardinal inequality involving homogeneous spaces

We give a short proof of van Douwen’s theorem that $|X| \leqslant {2^{\pi (X)}}$ holds for homogeneous Hausdorff spaces.

Nonhomogeneity of products of preimages and 𝜋-weight

We prove a general nonhomogeneity result which implies among others (1) if X is a homogeneous Hausdorff space, then | X | ⩽ 2 π ( X ) |X| \leqslant {2^{\pi (X)}} ; (2) no power of β ( ω ) − ω \beta (\omega ) - \omega , or of β Q − Q \beta Q - Q or of β R − R \beta R - R is homogeneous.

Conditions under which a connected representable space is locally connected

In this paper it is shown that every strongly locally homogeneous Hausdorff continuum is locally connected. It is known that every connected representable space is homogeneous and that every locally connected representable space is strongly locally homogeneous; this paper investigates the problem of whether or not every connected representable space is locally connected. The discovery that homogeneity did not characterize the circle among planar continua motivated other classifications of continua in terms of the actions of their homeomorphism groups, and the pseudoarc remains the most tortuous proving grounds for testing such topological properties. Those homogeneity-like properties that are not possessed by the pseudo-arc appear to be closely related to local connectedness. Such properties include strong local homogeneity, representability, (strong) 2-homogeneity and countable dense homogeneity. In this paper it is shown that every strongly locally homogeneous Hausdorff continuum is locally connected. It is known that every connected representable space is homogeneous and that every locally connected representable space is strongly locally homogeneous; this paper investigates the problem of whether or not every connected representable space is locally connected. It follows from results of Ben Fitzpatrick, Jr. and Ralph Bennett that every connected locally compact separable metric representable space is locally connected. Arguments of Fitzpatrick and de Groot are used to show that if (X, τ) is a representable connected complete metric space such that τ does not have a base of totally disconnected sets, then (X, τ) is locally connected (and hence arcwise connected and strongly locally homogeneous). The authors do not know of even a homogeneous connected complete metric space with a base of totally disconnected sets; however, there exists a separable connected metric topology that has such a base. Let (X, τ) be a topological space. We let H(X) denote the group of all homeomorphisms from the space (X, τ) onto itself and let i denote the identity of H(X). If A c X, then A' = {he H(X): h\A = i I A) and if G is a subgroup of H(X), then G' = {x e X: g(x) = x for each geG}. If GaH{X) and AcX, then G(A) = {g(a):geG and aeA}. We write G(x) in room of G({x}). Throughout this paper all spaces are assumed to be Hausdorff.