Van Mill Research Articles

Core compactness of ordered topological spaces

<abstract><p>In this paper, we investigate the property of core compactness of ordered topological spaces. Particularly, we give a series of characterizations of the core compactness for directed spaces. Several results obtained in this paper are closely related to a long-standing open problem in Open problems in Topology (J. van Mill, G. M. Reed Eds., North-Holland, 1990): Which distributive continuous lattice's spectrum is exactly a sober locally compact Scott space?</p></abstract>

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).

Dense metrizable subspaces in powers of Corson compacta

We characterize when the countable power of a Corson compactum has a dense metrizable subspace and construct consistent examples of Corson compacta whose countable power does not have a dense metrizable subspace. We also give several remarks about ccc Corson compacta and, as a byproduct, we obtain a new proof of Kunen and van Mill’s characterization of when a Corson compactum supporting a strictly positive measure is metrizable.

On the scope of the Effros theorem

All spaces (and groups) are assumed to be separable and metrizable. Jan van Mill showed that every analytic group $G$ is Effros (that is, every continuous transitive action of $G$ on a non-meager space is micro-transitive). We complete the picture by

A characterization of Erdős space factors

We prove that an almost zero-dimensional space $X$ is an Erd\H{o}s space factor if and only if $X$ has a Sierpi\'{n}ski stratification of C-sets. We apply this characterization to spaces which are countable unions of C-set Erd\H{o}s space factors. We show that the Erd\H{o}s space $\mathfrak E$ is unstable by giving strongly $\sigma$-complete and nowhere $\sigma$-complete examples of almost zero-dimensional $F_{\sigma\delta}$-spaces which are not Erd\H{o}s space factors. This answers a question by Dijkstra and van Mill.

The establishment and development of star-Lindelöf spaces

In 1983, Mumin Dai and Shôgo Ikenaga independently introduced mutually equivalent *Lindelöf spaces and ω-star spaces, later called strong 1-star-Lindelöf spaces in 1991, star-Lindelöf spaces in 1998 or star-countable spaces in 2007. After a series of work of E.K. van Douwen, M.V. Matveev and J. van Mill et al., star covering properties, the star-compactness and star-Lindelöfness being the cores, have become one of the central topics of general topology in nearly 20 years. In this paper, in clue of star-Lindelöf numbers, star covering properties and star-P properties, we review some research progress of star countable properties, introduce several unsolved problems and expound the contribution of Chinese scholars in this field.

On weakening tightness to weak tightness

The weak tightness wt(X) of a space X was introduced in Carlson (Topol Appl 249:103–111, 2018) with the property $$wt(X)\le t(X)$$. We investigate several well-known results concerning t(X) and consider whether they extend to the weak tightness setting. First we give an example of a non-sequential compactum X such that $$wt(X)=\aleph _0<t(X)$$ under $$2^{\aleph _0}=2^{\aleph _1}$$. In particular, this demonstrates the celebrated Balogh’s (Proc Am Math Soc 105(3):755–764, 1989) Theorem does not hold in general if countably tight is replaced with weakly countably tight. Second, we introduce the notion of an S-free sequence and show that if X is a homogeneous compactum then $$|X|\le 2^{wt(X)\pi \chi (X)}$$. This refines a theorem of de la Vega (Topol Appl 153:2118–2123, 2006). In the case where the cardinal invariants involved are countable, this also represents a variation of a theorem of Juhasz and van Mill (Proc Am Math Soc 146(1):429–437, 2018). In this connection we also show $$w(X)\le 2^{wt(X)}$$ for a homogeneous compactum. Third, we show that if X is a $$T_1$$ space, $$wt(X)\le \kappa $$, X is $$\kappa ^+$$-compact, and $$\psi (\overline{D},X)\le 2^\kappa $$ for any $$D\subseteq X$$ satisfying $$|D|\le 2^\kappa $$, then (a) $$d(X)\le 2^\kappa $$ and (b) X has at most $$2^\kappa $$-many $$G_\kappa $$-points. This is a variation of another theorem of Balogh (Topol Proc 27:9–14, 2003). Finally, we show that if X is a regular space, $$\kappa =L(X)wt(X)$$, and $$\lambda $$ is a caliber of X satisfying $$\kappa <\lambda \le \left( 2^{\kappa }\right) ^+$$, then $$d(X)\le 2^{\kappa }$$. This extends of theorem of Arhangel$$'$$skiĭ (Topol Appl 104:13–26, 2000).

ON THE CARDINALITY OF HOMOGENEOUS COMPACTA OF COUNTABLE TIGHTNESS

We prove that every homogeneous compacta of countable tightness and d(X)≤2 ℵ o , is first countable. A relevant conjecture is raised by Arhangel'skii, conjec- ture 1.17 in (1), see also van Mill (11), which says: every homogeneous compacta of countable tightness is first countable. For all undefined notions, see Engelking(6), Kunnen(10),and Juhasz(9). Recall that πχ(X), πχ(A ), πω(X), ω(X), d(X) and t(X) denote the π − character, π − character of A,π − weight, weight,density and tightness of X. A space X is homogeneous iff for every x,y ∈ X there is a homeomorphism f of X onto X with f(x) = y. A space is hereditarily separable (HS) iff every subspace is separable. A space is power homogeneous if X k is homogeneous for some k. All spaces under discussion are Tychonoff. In this paper, we prove that if a space X is homogeneous compactum of countable tightness and d(X)≤2 ℵ o , then it is first countable. Results of the same flavour were obtained by Bell (4),and Arhangel'skii (2). Bell proved that if X is a continuous image of a compact ordered space and X is power homogeneous, then X is first countable. Arhangel'skii proved that if X is Corson compact and power homogeneous then X is first countable, and a compact scattered power homogeneous space is countable. A recently interesting result was obtained by van Mill (12). He constructed a compactum of countable π − weight and character ℵ1 with the property that it is homogeneous under MA+� CH whereas CH implies that it is not.

On Spaces Star Determined by Compact Metrizable Subspaces

A space X is said to be star determined by compact metrizable subspaces (star-CM for short) if for any open cover $${\mathcal {U}}$$ of X there is a compact and metrizable subspace $$Y \subset X$$ such that $${\text {St}}(Y, {\mathcal {U}})=X$$ . This notation of star-CM was introduced by van Mill, Tkachuk and Wilson in (Topol Appl 154:2127–2134, 2007). In this paper, we investigate the relations between star-CM spaces and related spaces, and study topological properties of star-CM spaces. We also establish a cardinal theorem for star-CM spaces with symmetric g-functions.

Coloring Cantor sets and resolvability of pseudocompact spaces

Let us denote by $\Phi(\lambda,\mu)$ the statement that $\mathbb{B}(\lambda) = D(\lambda)^\omega$, i.e. the Baire space of weight $\lambda$, has a coloring with $\mu$ colors such that every homeomorphic copy of the Cantor set $\mathbb{C}$ in $\mathbb{B}(\lambda)$ picks up all the $\mu$ colors. We call a space $X$ $\pi$-regular if it is Hausdorff and for every nonempty open set $U$ in $X$ there is a nonempty open set $V$ such that $\overline{V} \subset U$. We recall that a space $X$ is called feebly compact if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact if and only if it is feebly compact. The main result of this paper is the following: Let $X$ be a crowded feebly compact $\pi$-regular space and $\mu$ be a fixed (finite or infinite) cardinal. If $\Phi(\lambda,\mu)$ holds for all $\lambda < \hat{c}(X)$ then $X$ is $\mu$-resolvable, i.e. $X$ contains $\mu$ pairwise disjoint dense subsets. (Here $\hat{c}(X)$ is the smallest cardinal $\kappa$ such that $X$ does not contain $\kappa$ many pairwise disjoint open sets.) This significantly improves earlier results of [van Mill J., {Every crowded pseudocompact ccc space is resolvable}, Topology Appl. 213 (2016), 127--134], or [Ortiz-Castillo Y. F., Tomita A. H., {Crowded pseudocompact Tychonoff spaces of cellularity at most the continuum are resolvable}, Conf. talk at Toposym 2016].

Cardinal Invariants for the $G_\delta $ topology

We prove upper bounds for the spread, the Lindelof number and the weak Lindelof number of the $G_\delta$-topology on a topological space and apply a few of our bounds to give a short proof to a recent result of Juhasz and van Mill regarding the cardinality of a $\sigma$-countably tight homogeneous compactum.

A study of chain conditions and dually properties

In this paper, we make some observations on chain conditions and dually properties. In particular, we show that:(1) A subspace X⊂ω1ω is dually CCC then e(X)≤ω and a normal subspace X⊂ω1ω is DCCC if and only if e(X)≤ω;(2) There is a Tychonoff pseudocompact subspace X⊂(ω1+1)2 which is not dually CCC;(3) In the class of o-semimetrizable spaces, dually separable is self-dual with respect to neighbourhood assignments. As an application, we obtain an example of a CCC normal Moore space which is not dually separable under MA+¬CH;(4) There exists an example of a large normal CCC semi-stratifiable space, which answers a question of Xuan and Song (2018) [21, Question 4.11];(5) Every dually separable and monotonically monolithic space is Lindelöf, which gives a partial answer to a question of Alas, Junqueira, van Mill, Tkachuk and Wilson (2011) [2, Question 2.1];(6) A dually separable Hausdorff space with a strong rank 1-diagonal has cardinality at most 2c. The conclusion is also true for regular spaces if we replace “strong rank 1-diagonal” with “Gδ-diagonal”;(7) A dually separable ω-monolithic Hausdorff space with a Gδ-diagonal has cardinality at most c.

On the weak tightness and power homogeneous compacta

Motivated by results of Juhász and van Mill in [13], we define the cardinal invariant wt(X), the weak tightness of a topological space X, and show that |X|≤2L(X)wt(X)ψ(X) for any Hausdorff space X (Theorem 2.8). As wt(X)≤t(X) for any space X, this generalizes the well-known cardinal inequality |X|≤2L(X)t(X)ψ(X) for Hausdorff spaces (Arhangel′skiĭ [1], Šapirovskiĭ [17]) in a new direction. Theorem 2.8 is generalized further using covers by Gκ-sets, where κ is a cardinal, to show that if X is a power homogeneous compactum with a countable cover of dense, countably tight subspaces then |X|≤c, the cardinality of the continuum. This extends a result in [13] to the power homogeneous setting.

TOWERS IN FILTERS, CARDINAL INVARIANTS, AND LUZIN TYPE FAMILIES

AbstractWe investigate which filters onωcan contain towers, that is, a modulo finite descending sequence without any pseudointersection (in${[\omega ]^\omega }$). We prove the following results:(1)Many classical examples of nice tall filters contain no towers (in ZFC).(2)It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well).(3)It is consistent that all towers generate nonmeager filters (this answers a question of P. Borodulin-Nadzieja and D. Chodounský), in particular (consistently) Borel filters do not contain towers.(4)The statement “Every ultrafilter contains towers.” is independent of ZFC (this improves an older result of K. Kunen, J. van Mill, and C. F. Mills).Furthermore, we study many possible logical (non)implications between the existence of towers in filters, inequalities between cardinal invariants of filters (${\rm{ad}}{{\rm{d}}^{\rm{*}}}\left( {\cal F} \right)$,${\rm{co}}{{\rm{f}}^{\rm{*}}}\left( {\cal F} \right)$,${\rm{no}}{{\rm{n}}^{\rm{*}}}\left( {\cal F} \right)$, and${\rm{co}}{{\rm{v}}^{\rm{*}}}\left( {\cal F} \right)$), and the existence of Luzin type families (of size$\ge {\omega _2}$), that is, if${\cal F}$is a filter then${\cal X} \subseteq {[\omega ]^\omega }$is an${\cal F}$-Luzin family if$\left\{ {X \in {\cal X}:|X \setminus F| = \omega } \right\}$is countable for every$F \in {\cal F}$.

Another proof of a theorem of van Mill and Wattel on weak selections

Another proof of a theorem of van Mill and Wattel on weak selections

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].

On topological groups with remainder of character k

&lt;p style="margin: 0px;"&gt;In [A.V. Arhangel'skii and J. van Mill, On topological groups with a first-countable remainder, Top. Proc. &lt;span id="OBJ_PREFIX_DWT1099_com_zimbra_phone" class="Object"&gt;42 (2013), 157-163&lt;/span&gt;] it is proved that the character of a non-locally compact topological group with a first countable remainder doesn't exceed $\omega_1$ and a non-locally compact topological group of character $\omega_1$ having a compactification whose reminder is first countable is given. We generalize these results in the general case of an arbitrary infinite cardinal k.&lt;/p&gt;&lt;p style="margin: 0px;"&gt; &lt;/p&gt;

Locally σ-compact rectifiable spaces

A topological space G is said to be a rectifiable space provided that there are a surjective homeomorphism φ:G×G→G×G and an element e∈G such that π1∘φ=π1 and for every x∈G, φ(x,x)=(x,e), where π1:G×G→G is the projection to the first coordinate. In this paper, we first prove that each locally compact rectifiable space is paracompact, which gives an affirmative answer to Arhangel'skii and Choban's question (Arhangel'skii and Choban [2]). Then we prove that every locally σ-compact rectifiable space with a bc-base is locally compact or zero-dimensional, which improves Arhangel'skii and van Mill's result (Arhangel'skii and van Mill [4]). Finally, we prove that each kω-rectifiable space is rectifiable complete.

Countable Dense Homogeneity in Powers of Zero-dimensional Definable Spaces

AbstractWe show that for a coanalytic subspace X of 2ω, the countable dense homogeneity of Xω is equivalent to X being Polish. This strengthens a result of Hruˇs´ak and Zamora Avilés. Then, inspired by results of Hernández-Guti´errez, Hruˇs´ak, and van Mill, using a technique of Medvedev, we construct a non-Polish subspace X of 2ω such that Xω is countable dense homogeneous. This gives the ûrst ZFC answer to a question of Hruˇs´ak and Zamora Avil´es. Furthermore, since our example is consistently analytic, the equivalence result mentioned above is sharp. Our results also answer a question of Medini and Milovich. Finally, we show that if every countable subset of a zero-dimensional separable metrizable space X is included in a Polish subspace of X, then Xω is countable dense homogeneous.

A remark on the invariance of dimension

Combining Kulpa’s proof of the cubical Sperner lemma and a dimension theoretic idea of van Mill we give a very short proof of the invariance of dimension, i.e. the statement that cubes $$[0,1]^n$$ and $$[0,1]^m$$ are homeomorphic if and only if n = m.