arXiv subjects

Ziqin Feng

Publications and source records attributed to Ziqin Feng.

18 recordsLinked to original sources

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 $\mathfrak{c} < \aleph_{ω_1}$, 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↗

arXiv2025

Vietoris-Rips complexes of torus grids

We study the topology of Vietoris--Rips complexes of finite grids on the torus. Let $T_{n,n}$ be the grid of $n\times n$ points on the flat torus $S^1\times S^1$, equipped with the $l^1$ metric. Let $\mathrm{VR}(T_{n,n};k)$ be the Vietoris--Rips simplicial complex of this torus grid at scale $k\ge 0$. For $n\ge 7$ and small scales $2\le k\le \frac{n-1}{3}$, the complex $\mathrm{VR}(T_{n,n};k)$ is homotopy equivalent to the torus. For large scales $k\ge 2\lfloor\frac{n}{2}\rfloor$, the complex $\mathrm{VR}(T_{n,n};k)$ is a simplex and hence contractible. Interesting topology arises over intermediate scales $\frac{n-1}{3}<k<2\lfloor\frac{n}{2}\rfloor$. For example, we prove that $\mathrm{VR}(T_{2n,2n};2n-1)\cong S^{2n^2-1}$ for $n\ge 2$, that $\mathrm{VR}(T_{3n,3n};n)\simeq\vee^{6n^2-1}S^2$ for $n\ge 2$, and that $\mathrm{VR}(T_{3n-1,3n-1};n)\simeq \bigvee_{6n-3} S^2\vee \bigvee_{6n-2}S^3$ for $n\geq 3$. Based on homology computations, we conjecture that $\mathrm{VR}(T_{n,n};k)$ is homotopy equivalent to a $3$-sphere for a countable family of $(n,k)$ pairs, and we prove this for $(n,k)=(7,4)$.

math.AT↗

arXiv2024

Facets in the Vietoris--Rips complexes of hypercubes

In this paper, we investigate the facets of the Vietoris--Rips complex $\mathcal{VR}(Q_n; r)$ where $Q_n$ denotes the $n$-dimensional hypercube. We are particularly interested in those facets which are somehow independent of the dimension $n$. Using Hadamard matrices, we prove that the number of different dimensions of such facets is a super-polynomial function of the scale $r$, assuming that $n$ is sufficiently large. We show also that the $(2r-1)$-th dimensional homology of the complex $\mathcal{VR}(Q_n; r)$ is non-trivial when $n$ is large enough, provided that the Hadamard matrix of order $2r$ exists.

math.AT↗

arXiv2024

Exploring Homological Properties of Independent Complexes of Kneser Graphs

We discuss the topological properties of the independence complex of Kneser graphs, Ind(KG$(n, k))$, with $n\geq 3$ and $k\geq 1$. By identifying one kind of maximal simplices through projective planes, we obtain homology generators for the $6$-dimensional homology of the complex Ind(KG$(3, k))$. Using cross-polytopal generators, we provide lower bounds for the rank of $p$-dimensional homology of the complex Ind(KG$(n, k))$ where $p=1/2\cdot {2n+k\choose 2n}$. Denote $\mathcal{F}_n^{[m]}$ to be the collection of $n$-subsets of $[m]$ equipped with the symmetric difference metric. We prove that if $\ell$ is the minimal integer with the $q$th dimensional reduced homology $\tilde{H}_q(\mathcal{VR}(\mathcal{F}^{[\ell]}_n; 2(n-1)))$ being non-trivial, then $$\text{rank} (\tilde{H}_q(\mathcal{VR}(\mathcal{F}_n^{[m]}; 2(n-1)))\geq \sum_{i=\ell}^m{i-2\choose \ell-2}\cdot \text{rank} (\tilde{H}_q(\mathcal{VR}(\mathcal{F}_n^{[\ell]}; 2(n-1))). $$ Since the independence complex Ind(KG$(n, k))$ and the Vietoris-Rips complex $\mathcal{VR}(\mathcal{F}^{[2n+k]}_n; 2(n-1))$ are the same, we obtain a homology propagation result in the setting of independence complexes of Kneser graphs. Connectivity of these complexes is also discussed in this paper.

math.CO↗

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 $ϕ:\mathcal{K}(X) \to \mathcal{K}(Y)$ 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 $σ$-pseudocompact or for some separable metrizable $M$, we have $\mathcal{K}(M) \ge_T (G,\mathcal{K}(G))$.

math.GN↗

arXiv2023

On Vietoris-Rips complexes of Finite Metric Spaces with Scale $2$

We examine the homotopy types of Vietoris-Rips complexes on certain finite metric spaces at scale $2$. We consider the collections of subsets of $[m]=\{1, 2, \ldots, m\}$ equipped with symmetric difference metric $d$, specifically, $\mathcal{F}^m_n$, $\mathcal{F}_n^m\cup \mathcal{F}^m_{n+1}$, $\mathcal{F}_n^m\cup \mathcal{F}^m_{n+2}$, and $\mathcal{F}_{\preceq A}^m$. Here $\mathcal{F}^m_n$ is the collection of size $n$ subsets of $[m]$ and $\mathcal{F}_{\preceq A}^m$ is the collection of subsets $\preceq A$ where $\preceq$ is a total order on the collections of subsets of $[m]$ and $A\subseteq [m]$ (see the definition of $\preceq$ in Section~\ref{Intro}). We prove that the Vietoris-Rips complexes $\mathcal{VR}(\mathcal{F}^m_n, 2)$ and $\mathcal{VR}(\mathcal{F}_n^m\cup \mathcal{F}^m_{n+1}, 2)$ are either contractible or homotopy equivalent to a wedge sum of $S^2$'s; also, the complexes $\mathcal{VR}(\mathcal{F}_n^m\cup \mathcal{F}^m_{n+2}, 2)$ and $\mathcal{VR}(\mathcal{F}_{\preceq A}^m, 2)$ are either contractible or homotopy equivalent to a wedge sum of $S^3$'s. We provide inductive formula for these homotopy types extending the result of Barmak in \cite{Bar13} about the independence complexes of Kneser graphs \text{KG}$_{2, k}$ and the result of Adamaszek and Adams in \cite{AA22} about Vietoris-Rips complexes of hypercube graphs with scale $2$.

math.CO↗

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↗

arXiv2023

Homotopy types of Vietoris-Rips complexes of Hypercube Graphs

We describe the homotopy types of Vietoris-Rips complexes of hypercube graphs at scale $3$. We represent the vertices in the hypercube graph $Q_m$ as the collection of all subsets of $[m]=\{1, 2, \ldots, m\}$ and equip $Q_m$ with the metric using symmetric difference distance. It is proved in \cite{AA22} that the Vietoris-Rips complexes of hypercube graphs $Q_m$ at scale $2$, $\mathcal{VR}(Q_m; 2)$, is homotopy equivalent to $c_m$-many spheres with dimension $3$ where $c_m=\sum_{0\leq j< i<m} (j+1)(2^{m-2}-2^{i-1})$. Questions are raised in \cite{AA22} for determining the homotopy types of $\mathcal{VR}(Q_m, r)$ with large scales $r=3, 4, \ldots, m-2$. We prove that for $m\geq 5$, $$\mathcal{VR}(Q_m; 3)\simeq (\bigvee_{2^{m-4}\cdot{m\choose 4}} S^7) \vee (\bigvee_{\sum_{i=4}^{m-1}2^{i-4}\cdot{i\choose 4}} S^4).$$

math.CO↗

arXiv2021

Sub-posets in $ω^ω$ and the Strong Pytkeev$^\ast$ Property

Tukey order are used to compare the cofinal complexity of partially order sets (posets). We prove that there is a $2^\mathfrak{c}$-sized collection of sub-posets in $2^ω$ which forms an antichain in the sense of Tukey ordering. Using the fact that any boundedly-complete sub-poset of $ω^ω$ is a Tukey quotient of $ω^ω$, we answer two open questions published in \cite{FKL16}. The relation between $P$-base and strong Pytkeev$^\ast$ property is investigated. Let $P$ be a poset equipped with a second-countable topology in which every convergent sequence is bounded. Then we prove that any topological space with a $P$-base has the strong Pytkeev$^\ast$ property. Furthermore, we prove that every uncountably-dimensional locally convex space (lcs) with a $P$-base contains an infinite-dimensional metrizable compact subspace. Examples in function spaces are given.

math.GN↗

arXiv2021

Compact Spaces with a $P$-base

In the paper, we investigate (scattered) compact spaces with a $P$-base for some poset $P$. More specifically, we prove that, under the assumption $ω_1<\mathfrak{b}$, any compact space with an $ω^ω$-base is first-countable and any scattered compact space with an $ω^ω$-base is countable. These give positive solutions to Problems 8.6.9 and 8.7.7 in \cite{Banakh2019}. Using forcing, we also prove that in a model of $ω_1<\mathfrak{b}$, there is a non-first-countable compact space with a $P$-base for some poset $P$ with calibre~$ω_1$.

math.GN↗

arXiv2019

Continuous Selections of Lower semicontinuous Set-valued Mappings

A space $X$ is strongly $Y$-selective (resp., $Y$-selective) if every lower semicontinuous mapping from $Y$ to the nonempty subsets (resp., nonempty closed subsets) of $X$ has a continuous selection. We also call $X$ (strongly) $C$-selective if it is (strongly) $Y$-selective for any countable space $Y$, and (strongly) $L$-selective if it is (strongly) ($ω+1$)-selective. E. Michael showed that every first countable space is strongly $C$-selective. We extend this by showing that every $W$-space in the sense of the second author is strongly $C$-selective. We also show that every GO-space is $C$-selective, and that every $L$-selective space has Arhangel'skii's property $α_1$. We obtain an example under $\mathfrak p=\mathfrak c$ of a strongly $L$-selective space that is not $C$-selective, and we show that it is consistent with and independent of ZFC that a space is strongly $L$-selective iff it is $L$-selective and Fréchet. Finally, we answer a question of the third author and Junnila by showing that the ordinal space $ω_1 +1$ is not self-selective.

math.GN↗

arXiv2018

Spaces with a $\mathbb{Q}$-diagonal

A space $X$ has a $\mathbb{Q}$-diagonal if $X^2\setminus Δ$ has a $\mathcal{K}(\mathbb{Q})$-directed compact cover. We show that any compact space with a $\mathbb{Q}$-diagonal is metrizable, hence any Tychonorff space with a $\mathbb{Q}$-diagonal is cosmic. These give a positive answer to Problem 4.2 and Problem 4.8 in \cite{COT11} raised by Cascales, Orihuela and Tkachuk.

math.GN↗

arXiv2018

A note on Collins-Roscoe Structuring Mechanism

A space $X$ has countable $(F)$-property if it has countable point network satisfying the Collins-Roscoe structuring mechanism. Some sufficient conditions for $C_p(X)$ having countable $(F)$-property are obtained. As a corollary, we prove that if $X$ is Corson compact, $C_p(X)$ satisfies countable $(F)$. This answers a question raised by Tkachuk. Also we get a class of function spaces with hereditarily $D$-property. We also prove that the countable $(F)$-property is preserved by taking $Σ_s$-product. This answers questions of Tkachuk positively.

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

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↗

arXiv2009

On Hilbert's 13th Problem

Every continuous function of two or more real variables can be written as the superposition of continuous functions of one real variable along with addition.

math.CA↗

arXiv2009

Minimal Size of Basic Families

A family $\bfam$ of continuous real-valued functions on a space $X$ is said to be {\sl basic} if every $f \in C(X)$ can be represented $f = \sum_{i=1}^n g_i \circ ϕ_i$ for some $ϕ_i \in \bfam$ and $g_i \in C(\R)$ ($i=1, ..., n$). Define $\basic (X) = \min \{|\bfam| : \bfam$ is a basic family for $X\}$. If $X$ is separable metrizable $X$ then either $X$ is locally compact and finite dimensional, and $\basic (X) < \aleph_0$, or $\basic (X) = \mathfrak{c}$. If $K$ is compact and either $w(K)$ (the minimal size of a basis for $K$) has uncountable cofinality or $K$ has a discrete subset $D$ with $|D|=w(K)$ then either $K$ is finite dimensional, and $\basic (K) = \cof ([w(K)]^{\aleph_0}, \subseteq)$, or $\basic (K) = |C(K)|=w(K)^{\aleph_0}$.

math.GN↗