We prove that for every compactum X and every integer n ≥ 2 there are a compactum Z of dimension ≤ n+1 and a surjective $U{V}^{n-1}$-map r: Z → X such that for every abelian group G and every integer k ≥ 2 such that $di{m}_{G}X\le k\le n$ we have $di{m}_{G}Z\le k$ and r is G-acyclic.

We prove that $trt{K}_{\omega }>\omega +1$, where trt stands for the transfinite extension of Steinke’s separation dimension. This answers a question of Chatyrko and Hattori.

Let f: X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g: X → ⁿ with dim(f △ g) = 0 is uniformly dense in C(X,ⁿ); (2) for every 0 ≤ k ≤ n-1 there exists an $F_{\sigma }$-subset $A_k$ of X such that $dim{A}_k\le k$ and the restriction $f\left|\right(X\setminus A_k)$ is (n-k-1)-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij...

Let $G$ be a finite abelian group with identity element $1_G$ and $L={⨁}_{g\in G}{L}^g$ be an infinite dimensional $G$-homogeneous vector space over a field of characteristic $0$. Let $E=E\left(L\right)$ be the Grassmann algebra generated by $L$. It follows that $E$ is a $G$-graded algebra. Let $\left|G\right|$ be odd, then we prove that in order to describe any ideal of $G$-graded identities of $E$ it is sufficient to deal with $G^{\text{'}}$-grading, where $|G^{\text{'}}|\le |G|$, ${dim}_F{L}^{1_{G^{\text{'}}}}=\infty$ and ${dim}_F{L}^{g^{\text{'}}}<\infty$ if $g^{\text{'}}\ne 1_{G^{\text{'}}}$. In the same spirit of the case $\left|G\right|$ odd, if $\left|G\right|$ is even it is sufficient to study only those $G$-gradings...

We say that a collection $𝒜$ of subsets of $X$ has property $\left(CC\right)$ if there is a set $D$ and point-countable collections $𝒞$ of closed subsets of $X$ such that for any $A\in 𝒜$ there is a finite subcollection $ℱ$ of $𝒞$ such that $A=D\setminus \bigcup ℱ$. Then we prove that any compact space is Corson if and only if it has a point-$\sigma$-$\left(CC\right)$ base. A characterization of Corson compacta in terms of (strong) point network is also given. This provides an answer to an open question in “A Biased View of Topology as a Tool in Functional Analysis”...

We classify classical linear connections $A(\Gamma ,\Lambda ,\Theta )$ on the total space $Y$ of a fibred manifold $Y\to M$ induced in a natural way by the following three objects: a general connection $\Gamma$ in $Y\to M$, a classical linear connection $\Lambda$ on $M$ and a linear connection $\Theta$ in the vertical bundle $VY\to Y$. The main result says that if $\dim \left(M\right)\ge 3$ and $\dim \left(Y\right)-\dim \left(M\right)\ge 3$ then the natural operators $A$ under consideration form the $17$ dimensional affine space.

We characterize, in terms of X, the extensional dimension of the Stone-Čech corona βX∖X of a locally compact and Lindelöf space X. The non-Lindelöf case is also settled in terms of extending proper maps with values in $I^{\tau }\setminus L$, where L is a finite complex. Further, for a finite complex L, an uncountable cardinal τ and a $Z_{\tau }$-set X in the Tikhonov cube $I^{\tau }$ we find a necessary and sufficient condition, in terms of $I^{\tau }\setminus X$, for X to be in the class AE([L]). We also introduce a concept of a proper absolute...

It is shown that there exists a $\sigma$-compact topological group which cannot be represented as a continuous image of a Lindelöf $p$-group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf $p$-groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space $Y$ is a continuous image of a Lindelöf $p$-group, then there exists a covering $\gamma$ of $Y$ by dyadic compacta such that $\left|\gamma \right|\le 2^{\omega }$. We also show that if a homogeneous compact space $Y$ is...

Let $R$ be a commutative ring with unity. The notion of maximal non valuation domain in an integral domain is introduced and characterized. A proper subring $R$ of an integral domain $S$ is called a maximal non valuation domain in $S$ if $R$ is not a valuation subring of $S$, and for any ring $T$ such that $R\subset T\subset S$, $T$ is a valuation subring of $S$. For a local domain $S$, the equivalence of an integrally closed maximal non VD in $S$ and a maximal non local subring of $S$ is established. The relation between $dim(R,S)$ and...

Let $\Delta$ be a pure simplicial complex on the vertex set $\left[n\right]=\{1,...,n\}$ and $I_{\Delta }$ its Stanley-Reisner ideal in the polynomial ring $S=K[x_1,...,x_n]$. We show that $\Delta$ is a matroid (complete intersection) if and only if $S/I_{\Delta }^{\left(m\right)}$ ($S/I_{\Delta }^m$) is clean for all $m\in \mathbb{N}$ and this is equivalent to saying that $S/I_{\Delta }^{\left(m\right)}$ ($S/I_{\Delta }^m$, respectively) is Cohen-Macaulay for all $m\in \mathbb{N}$. By this result, we show that there exists a monomial ideal $I$ with (pretty) cleanness property while $S/I^m$ or $S/I^{\left(m\right)}$ is not (pretty) clean for all integer $m\ge 3$. If $dim\left(\Delta \right)=1$, we also prove that $S/I_{\Delta }^{\left(2\right)}$ ($S/I_{\Delta }^2$) is clean if and only...

We investigate how one can detect the dualizing property for a chain complex over a commutative local Noetherian ring $R$. Our focus is on homological properties of contracting endomorphisms of $R$, e.g., the Frobenius endomorphism when $R$ contains a field of positive characteristic. For instance, in this case, when $R$ is $F$-finite and $C$ is a semidualizing $R$-complex, we prove that the following conditions are equivalent: (i) $C$ is a dualizing $R$-complex; (ii) $C\sim 𝐑{\mathrm{Hom}}_R{(}^{n}R,C)$ for some $n>0$; (iii) ${\mathrm{G}}_C{\text{-dim}}^{n}R<\infty$ and $C$ is derived...

Given d ≥ 2 consider the family of polynomials $P_c\left(z\right)=z^d+c$ for c ∈ ℂ. Denote by $J_c$ the Julia set of $P_c$ and let ${ℳ}_d=c|J_{c}isconnected$ be the connectedness locus; for d = 2 it is called the Mandelbrot set. We study semihyperbolic parameters $c₀\in \partial {ℳ}_d$: those for which the critical point 0 is not recurrent by $P_{c₀}$ and without parabolic cycles. The Hausdorff dimension of $J_c$, denoted by $HD\left(J_c\right)$, does not depend continuously on c at such $c₀\in \partial {ℳ}_d$; on the other hand the function $c\mapsto HD\left(J_c\right)$ is analytic in $ℂ-{ℳ}_d$. Our first result asserts that there is still some...

Let $𝔞\subseteq ℂ[x_1,...,x_n]$ be a monomial ideal and $𝒥\left({𝔞}^c\right)$ the multiplier ideal of $𝔞$ with coefficient $c$. Then $𝒥\left({𝔞}^c\right)$ is also a monomial ideal of $ℂ[x_1,...,x_n]$, and the equality $𝒥\left({𝔞}^c\right)=𝔞$ implies that $0<c<n+1$. We mainly discuss the problem when $𝒥\left(𝔞\right)=𝔞$ or $𝒥\left({𝔞}^{n+1-\epsilon }\right)=𝔞$ for all $0<\epsilon <1$. It is proved that if $𝒥\left(𝔞\right)=𝔞$ then $𝔞$ is principal, and if $𝒥\left({𝔞}^{n+1-\epsilon }\right)=𝔞$ holds for all $0<\epsilon <1$ then $𝔞=(x_1,...,x_n)$. One global result is also obtained. Let $\widetilde{𝔞}$ be the ideal sheaf on ${\mathbb{P}}^{n-1}$ associated with $𝔞$. Then it is proved that the equality $𝒥\left(\widetilde{𝔞}\right)=\widetilde{𝔞}$ implies that $\widetilde{𝔞}$ is principal.

The first author has recently proved that if f: X → Y is a k-dimensional map between compacta and Y is p-dimensional (0 ≤ k, p < ∞), then for each 0 ≤ i ≤ p + k, the set of maps g in the space $C(X,I^{p+2k+1-i})$ such that the diagonal product $f\times g:X\to Y\times I^{p+2k+1-i}$ is an (i+1)-to-1 map is a dense $G_{\delta }$-subset of $C(X,I^{p+2k+1-i})$. In this paper, we prove that if f: X → Y is as above and $D_j$ (j = 1,..., k) are superdendrites, then the set of maps h in $C(X,{\prod }_{j=1}^k{D}_j\times I^{p+1-i})$ such that $f\times h:X\to Y\times \left({\prod }_{j=1}^k{D}_j\times I^{p+1-i}\right)$ is (i+1)-to-1 is a dense $G_{\delta }$-subset of $C(X,{\prod }_{j=1}^k{D}_j\times I^{p+1-i})$ for each 0 ≤ i ≤ p.

Let $L_c\left(X\right)=\{f\in C\left(X\right):\overline{C_f}=X\}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $\left|f\left(U\right)\right|\le {\aleph }_0$. In this paper, we present a pointfree topology version of $L_c\left(X\right)$, named ${ℛ}_{\ell c}\left(L\right)$. We observe that ${ℛ}_{\ell c}\left(L\right)$ enjoys most of the important properties shared by $ℛ\left(L\right)$ and ${ℛ}_c\left(L\right)$, where ${ℛ}_c\left(L\right)$ is the pointfree version of all continuous functions of $C\left(X\right)$ with countable image. The interrelation between $ℛ\left(L\right)$, ${ℛ}_{\ell c}\left(L\right)$, and ${ℛ}_c\left(L\right)$ is examined. We show that $L_c\left(X\right)\cong {ℛ}_{\ell c}\left(𝔒\left(X\right)\right)$ for any space $X$. Frames $L$ for which ${ℛ}_{\ell c}\left(L\right)=ℛ\left(L\right)$ are characterized.