arXiv subjects

Paul Gartside

Publications and source records attributed to Paul Gartside.

At least 19 recordsLinked to original sources

arXiv2026

The Cofinality of Generating Familes

The topology of a separable metrizable space $M$ is \emph{generated} by a family $\mathcal{C}$ of its subsets provided that a set $A\subseteq M$ is closed in $M$ if and only if $A\cap C$ is closed in $C$ for each $C\in \mathcal{C}$. The \emph{sequentiality number}, $\mathop{seq}(M)$, and \emph{$k$-ness number}, $\mathop{k}(M)$, of $M$, are the minimum size of a generating family of convergent sequences, respectively compact subsets. Let $\mathfrak{b}$ be the minimum size of an unbounded set in $\omega^\omega$ with the mod finite order. For a cardinal $\kappa$, the \emph{covering number}, $\mathop{cov}(\kappa)$, is the minimum size of a family of countable subsets of $\kappa$ so that every countable subset of $\kappa$ is contained in an element of the family. It is shown using the Tukey order on relations that (1) $\mathop{seq}(M)=\mathop{cov}(|M|)\cdot \mathfrak{b}$, unless $M$ is locally small (every point of $M$ has a neighborhood of size strictly less than $|M|$) in which case $\mathop{seq}(M)=\lim_{\mu <|M|} \mathop{cov}(\mu)\cdot \mathfrak{b}$ and (2) $k(M)$ is in the interval $[kc(M)\cdot\mathfrak{b},\mathop{cov}(kc(M))\cdot \mathfrak{b}]$, where $kc(M)$ is the minimum number of compact sets that cover $M$. Solutions to problems of van Douwen's on the $k$-ness number of analytic and of co-analytic spaces are deduced.

math.LO↗

arXiv2025

The Shape of Generating Families

The topology of a space $X$ is generated by a family $\mathcal{C}$ of its subsets provided that a set $A\subseteq X$ is closed in $X$ if and only if $A\cap C$ is closed in $C$ for each $C\in \mathcal{C}$. A space $X$ is a $k$-space (respectively, sequential) if its topology is generated by the collection of all compact subsets (respectively, convergent sequences) of $X$. Relations are defined to capture the notion of a space being a $k$-space or sequential. The structure (or `shape') under the Tukey order of these relations applied to separable metrizable spaces is examined. For the $k$-space case the initial structure is completely determined, and the cofinal structure is shown to be highly complex. In the sequential case, however, the entire shape is determined. It follows that the number of Tukey types in the sequential case lies between $\aleph_0$ and $\mathfrak{c}$, is equal to $\aleph_0$ precisely when $\mathfrak{c} < \aleph_{\omega_1}$, and is equal to $\mathfrak{c}$ if and only if $\mathfrak{c}$ is a fixed point of the aleph function, necessarily of uncountable cofinality.

math.GN↗

arXiv2024

Products of Directed Sets with Calibre $(ω_1, ω)$

A directed set $P$ is calibre $(ω_1, ω)$ if every uncountable subset of $P$ contains an infinite bounded subset. $P$ is productively calibre $(ω_1, ω)$ if $P \times Q$ is calibre $(ω_1, ω)$ for every directed set $Q$ with calibre $(ω_1, ω)$, and $P$ is powerfully calibre $(ω_1, ω)$ if the countable power of $P$ is calibre $(ω_1, ω)$. It is shown that (1) uncountable products are calibre $(ω_1, ω)$ only in highly restrictive circumstances, (2) many but not all $\sum$-products of calibre $(ω_1, ω)$ directed sets are calibre $(ω_1, ω)$, (3) there are directed sets which are calibre $(ω_1, ω)$ but neither productively nor powerfully calibre $(ω_1, ω)$, and (4) there are directed sets which are powerfully but not productively calibre $(ω_1, ω)$. As an application, the position is established of $\sum ω^{ω_1}$ in the Tukey order among Isbell's classical 10 directed sets.

math.LO↗

arXiv2024

The Shape of Compact Covers

For a space $X$ let $\mathcal{K}(X)$ be the set of compact subsets of $X$ ordered by inclusion. A map $\phi:\mathcal{K}(X) \to \mathcal{K}(Y)$ is a relative Tukey quotient if it carries compact covers to compact covers. When there is such a Tukey quotient write $(X,\mathcal{K}(X)) \ge_T (Y,\mathcal{K}(Y))$, and write $(X,\mathcal{K}(X)) =_T (Y,\mathcal{K}(Y))$ if $(X,\mathcal{K}(X)) \ge_T (Y,\mathcal{K}(Y))$ and vice versa. We investigate the initial structure of pairs $(X,\mathcal{K}(X))$ under the relative Tukey order, focusing on the case of separable metrizable spaces. Connections are made to Menger spaces. Applications are given demonstrating the diversity of free topological groups, and related free objects, over separable metrizable spaces. It is shown a topological group $G$ has the countable chain condition if it is either $\sigma$-pseudocompact or for some separable metrizable $M$, we have $\mathcal{K}(M) \ge_T (G,\mathcal{K}(G))$.

math.GN↗

arXiv2023

Directed Sets of Topology -- Tukey Representation and Rejection

Every directed set is Tukey equivalent to (a) the family of all compact subsets, ordered by inclusion, of a (locally compact) space, to (b) a neighborhood filter, ordered by reverse inclusion, of a point (of a compact space, and of a topological group), and to (c) the universal uniformity, ordered by reverse inclusion, of a space. Two directed sets are Tukey equivalent if they are cofinally equivalent in the sense that they can both be order embedded cofinally in a third directed set. In contrast, any totally bounded uniformity is Tukey equivalent to $[κ]^{<ω}$, the collection of all finite subsets of $κ$, where $κ$ is the cofinality of the uniformity. All other Tukey types are `rejected' by totally bounded uniformities. Equivalently, a compact space $X$ has weight (minimal size of a base) equal to $κ$ if and only if the neighborhood filter of the diagonal is Tukey equivalent to $[κ]^{<ω}$. A number of questions from the literature are answered with the aid of the above results.

math.GN↗

arXiv2021

Eulerian Spaces

We develop a unified theory of Eulerian spaces by combining the combinatorial theory of infinite, locally finite Eulerian graphs as introduced by Diestel and Kühn with the topological theory of Eulerian continua defined as irreducible images of the circle, as proposed by Bula, Nikiel and Tymchatyn. First, we clarify the notion of an Eulerian space and establish that all competing definitions in the literature are in fact equivalent. Next, responding to an unsolved problem of Treybig and Ward from 1981, we formulate a combinatorial conjecture for characterising the Eulerian spaces, in a manner that naturally extends the characterisation for finite Eulerian graphs. Finally, we present far-reaching results in support of our conjecture which together subsume and extend all known results about the Eulerianity of infinite graphs and continua to date. In particular, we characterise all one-dimensional Eulerian spaces.

math.GN↗

arXiv2018

$n$-arc and $n$-circle connected graph-like spaces

A space $X$ is $n$-arc connected (respectively, $n$-circle connected) if for any choice of at most $n$ points there is an arc (respectively, a circle) in $X$ containing the specified points. We study $n$-arc connectedness and $n$-circle connectedness in compactifications of locally finite graphs and the slightly more general class of graph-like continua, uncovering a striking difference in their behaviour regarding $n$-arc and -circle connectedness.

math.CO↗

arXiv2018

n-Arc Connected Graphs

Given a graph G, of arbitrary size and unbounded vertex degree, denote by |G| the one-complex associated with $G$. The topological space |G| is n-arc connected (n-ac) if every set of no more than n points of |G| are contained in an arc (a homeomorphic copy of the closed unit interval). For any graph G, we show the following are equivalent: (i) |G| in 7-ac, (ii) |G| is n-ac for all n, and (iii) G is a subdivision of one of nine graphs. A graph G has |G| 6-ac if and only if either G is one of the nine 7-ac graphs, or, after suppressing all degree-2-vertices, the graph G is 3-regular, 3-connected, and removing any 6 edges does not disconnect G into 4 or more components. Similar combinatorial characterizations of graphs G such that |G| is n-ac for n=3, 4 and 5 are given. Together these results yield a complete classification of n-ac graphs, for all n.

math.CO↗

arXiv2016

Tukey Order, Calibres and the Rationals

One partially ordered set, $Q$, is a Tukey quotient of another, $P$, denoted $P \geq_T Q$, if there is a map $ϕ: P \to Q$ carrying cofinal sets of $P$ to cofinal sets of $Q$. Let $X$ be a space and denote by $\mathcal{K}(X)$ the set of compact subsets of $X$, ordered by inclusion. For certain separable metrizable spaces $M$, Tukey upper and lower bounds of $\mathcal{K}(M)$ are calculated. Results on invariants of $\mathcal{K}(M)$'s are deduced. The structure of all $\mathcal{K}(M)$'s under $\le_T$ is investigated. Particular emphasis is placed on the position of $\mathcal{K}(M)$ when $M$ is: completely metrizable, the rationals $\mathbb{Q}$, co-analytic or analytic.

math.GN↗

arXiv2016

Graph-Like Compacta: Characterizations and Eulerian Loops

A compact graph-like space is a triple $(X,V,E)$ where $X$ is a compact, metrizable space, $V \subseteq X$ is a closed zero-dimensional subset, and $E$ is an index set such that $X \setminus V \cong E \times (0,1)$. New characterizations of compact graph-like spaces are given, connecting them to certain classes of continua, and to standard subspaces of Freudenthal compactifications of locally finite graphs. These are applied to characterize Eulerian graph-like compacta.

math.CO↗

arXiv2016

The Tukey Order and Subsets of $ω_1$

One partially ordered set, $Q$, is a Tukey quotient of another, $P$, if there is a map $ϕ: P \to Q$ carrying cofinal sets of $P$ to cofinal sets of $Q$. Two partial orders which are mutual Tukey quotients are said to be Tukey equivalent. Let $X$ be a space and denote by $\mathcal{K}(X)$ the set of compact subsets of $X$, ordered by inclusion. The principal object of this paper is to analyze the Tukey equivalence classes of $\mathcal{K}(S)$ corresponding to various subspaces $S$ of $ω_1$, their Tukey invariants, and hence the Tukey relations between them. It is shown that $ω^ω$ is a strict Tukey quotient of $Σ(ω^{ω_1})$ and thus we distinguish between two Tukey classes out of Isbell's ten partially ordered sets. The relationships between Tukey equivalence classes of $\mathcal{K}(S)$, where $S$ is a subspace of $ω_1$, and $\mathcal{K}(M)$, where $M$ is a separable metrizable space, are revealed. Applications are given to function spaces.

math.GN↗

arXiv2015

Minimum Topological Group Topologies

A Hausdorff topological group topology on a group $G$ is the minimum (Hausdorff) group topology if it is contained in every Hausdorff group topology on $G$. For every compact metrizable space $X$ containing an open $n$-cell, $n\ge2$, the homeomorphism group $H(X)$ has no minimum Hausdorff group topology. The homeomorphism groups of the Cantor set and the Hilbert cube have no minimum group topology. For every compact metrizable space $X$ containing a dense open one-manifold, $H(X)$ has the minimum group topology. Some, but not all, oligomorphic groups have the minimum group topology.

math.GN↗

arXiv2015

Spaces l-Dominated by I or R

If $X$ is compact metrizable and has finite fd-height then the unit interval, $I$, $\ell$-dominates $X$, in other words, there is a continuous linear map of $C_p(I)$ onto $C_p(X)$. If the unit interval $\ell$-dominates a space $X$ then $X$ is compact metrizable and has countable fd-height. Similar results are given for spaces $\ell$-dominated by the reals.

math.GN↗

arXiv2015

Reconstructing Compact Metrizable Spaces

The deck, $\mathcal{D}(X)$, of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}]\colon x \in X\}$, where $[Y]$ denotes the homeomorphism class of $Y$. A space $X$ is (topologically) reconstructible if whenever $\mathcal{D}(Z)=\mathcal{D}(X)$ then $Z$ is homeomorphic to $X$. It is known that every (metrizable) continuum is reconstructible, whereas the Cantor set is non-reconstructible. The main result of this paper characterises the non-reconstructible compact metrizable spaces as precisely those where for each point $x$ there is a sequence $\langle B_n^x \colon n \in \mathbb{N}\rangle$ of pairwise disjoint clopen subsets converging to $x$ such that $B_n^x$ and $B_n^y$ are homeomorphic for each $n$, and all $x$ and $y$. In a non-reconstructible compact metrizable space the set of $1$-point components forms a dense $G_δ$. For $h$-homogeneous spaces, this condition is sufficient for non-reconstruction. A wide variety of spaces with a dense $G_δ$ set of $1$-point components are presented, some reconstructible and others not reconstructible.

math.GN↗

arXiv2015

Reconstructing Topological Graphs and Continua

The deck of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}] \colon x \in X\}$, where $[Z]$ denotes the homeomorphism class of $Z$. A space $X$ is topologically reconstructible if whenever $\mathcal{D}(X)=\mathcal{D}(Y)$ then $X$ is homeomorphic to $Y$. It is shown that all metrizable compact connected spaces are reconstructible. It follows that all finite graphs, when viewed as a 1-dimensional cell-complex, are reconstructible in the topological sense, and more generally, that all compact graph-like spaces are reconstructible.

math.GN↗

arXiv2015

Point Networks for Special Subspaces of $\mathbb{R}^κ$

Uniform characterizations of certain special subspaces of products of lines are presented. The characterizations all involve a collection of subsets (base, almost subbase, network or point network) organized by a directed set. New characterizations of Eberlein, Talagrand and Gulko compacta follow.

math.GN↗

arXiv2015

$P$-Paracompact and $P$-Metrizable Spaces

Let $P$ be a directed set and $X$ a space. A collection $\mathcal{C}$ of subsets of $X$ is \emph{$P$-locally finite} if $\mathcal{C}=\bigcup \{ \mathcal{C}_p : p \in P\}$ where (i) if $p \le p'$ then $\mathcal{C}_p \subseteq \mathcal{C}_{p'}$ and (ii) each $\mathcal{C}_p$ is locally finite. Then $X$ is \emph{$P$-paracompact} if every open cover has a $P$-locally finite open refinement. Further, $X$ is \emph{$P$-metrizable} if it has a $(P \times \mathbb{N})$-locally finite base. This work provides the first detailed study of $P$-paracompact and $P$-metrizable spaces, particularly in the case when $P$ is a $\mathcal{K}(M)$, the set of all compact subsets of a separable metrizable space $M$ ordered by set inclusion.

math.GN↗

arXiv2012

n-Arc Connected Spaces

A space is `n-arc connected' (n-ac) if any family of no more than n-points are contained in an arc. For graphs the following are equivalent: (i) 7-ac, (ii) n-ac for all n, (iii) continuous injective image of a closed sub-interval of the real line, and (iv) one of a finite family of graphs. General continua that are aleph_0-ac are characterized. The complexity of characterizing n-ac graphs for n=2,3,4,5 is determined to be strictly higher than that of the stated characterization of 7-ac graphs.

math.GN↗