Related papers: A note on Noetherian type of spaces
The cardinal invariant "Noetherian type" of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and…
A base of a topological space is called {\em Noetherian } iff it does not contain an infinite strictly $\subseteq$-increasing chain. We show that minimal cardinality of a regular spaces without a Noetherian base is the first strongly…
The Noetherian type of a space is the least $\kappa$ such that it has a base that is $\kappa$-like with respect to containment. Just as all known homogeneous compacta have cellularity at most $2^\omega$, they satisfy similar upper bounds in…
Extending some results of Malykhin, we prove several independence results about base properties of $\beta\omega-\omega$ and its powers, especially the Noetherian type $Nt(\beta\omega-\omega)$, the least $\kappa$ for which…
The Noetherian type of a space is the least k for which the space has a k^op-like base, i.e., a base in which no element has k-many supersets. We prove some results about Noetherian types of (generalized) ordered spaces and products…
A ring $R$ is called left strictly $(<\aleph_{\alpha})$-noetherian if $\aleph_{\alpha}$ is the minimum cardinal such that every ideal of $R$ is $(<\aleph_{\alpha})$-generated. In this note, we show that for every singular (resp., regular)…
A topological space $X$ is a $\Delta$-space (or $X \in \Delta$) if for any decreasing sequence $\{A_n : n < \omega\}$ of subsets of $X$ with empty intersection there is a (decreasing) sequence $\{U_n : n < \omega\}$ of open sets with empty…
Noetherian type and Noetherian $\pi$-type are two cardinal functions which were introduced by Peregudov in 1997, capturing some properties studied earlier by the Russian School. Their behavior has been shown to be akin to that of the…
A first-order theory is Noetherian with respect to the collection of formulae $\mathcal{F}$ if every definable set is a Boolean combination of instances of formulae in $\mathcal{F}$ and the topology whose subbasis of closed sets is the…
In the presence of suitable power spaces, compactness of $\mathbf{X}$ can be characterized as the singleton $\{X\}$ being open in the space $\mathcal{O}(\mathbf{X})$ of open subsets of $\mathbf{X}$. Equivalently, this means that universal…
A compact topological space X is spectral if it is sober (i.e., every irreducible closed set is the closure of a unique singleton) and the compact open subsets of X form a basis of the topology of X, closed under finite intersections.…
A space $X$ is said to be $\kappa$-resolvable (resp. almost $\kappa$-resolvable) if it contains $\kappa$ dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets). $X$ is maximally resolvable iff…
We investigate Noetherian families and show that every topological space has a Noetherian $\pi$-base. We prove that if a topological space has some special Noetherian $\pi$-bases, then NONEMPTY has a 2-tactic in the Banach-Mazur game on a…
A space X is called an alpha-Toronto space if X is scattered of Cantor-Bendixson rank alpha and is homeomorphic to each of its subspaces of same rank. We answer a question of Steprans by constructing a countable alpha-Toronto space for each…
The pinning down number $ {pd}(X)$ of a topological space $X$ is the smallest cardinal $\kappa$ such that for any neighborhood assignment $U:X\to \tau_X$ there is a set $A\in [X]^\kappa$ with $A\cap U(x)\ne\emptyset$ for all $x\in X$.…
We characterize when the Zariski space $\mathrm{Zar}(K|D)$ (where $D$ is an integral domain, $K$ is a field containing $D$ and $D$ is integrally closed in $K$) and the set $\mathrm{Zar_{min}}(L|D)$ of its minimal elements are Noetherian…
There are many examples of the fact that dimension and codimension behave somewhat counterintuitively. In EGA it is stated that a topological space is equidimensional, equicodimensional and catenary if and only if every maximal chain of…
Let $\mathcal M_X$ denote the ideal of meager subsets of a topological space $X$. We prove that if $X$ is a completely metrizable space without isolated points, then the smallest cardinality of a non-meager subset of $X$, denoted…
The main result of this note is the following theorem. "If $X$ is any Hausdorff space with $\kappa = \widehat{F}(X) \cdot \widehat{\mu}(X)$ then $L(X_{< \kappa}) \le \varrho(\kappa)$". Here $\widehat{F}(X)$ is the smallest cardinal…
The class of loop spaces whose mod p cohomology is Noetherian is much larger than the class of p-compact groups (for which the mod p cohomology is required to be finite). It contains Eilenberg-Mac Lane spaces such as the infinite complex…