### Struktur- und Anzahlformeln für Topologien auf endlichen Mengen.

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Given a set $X$ of “indeterminates” and a field $F$, an ideal in the polynomial ring $R=F\left[X\right]$ is called conservative if it contains with any polynomial all of its monomials. The map $S\mapsto RS$ yields an isomorphism between the power set $\text{P}\phantom{\rule{0.166667em}{0ex}}\left(X\right)$ and the complete lattice of all conservative prime ideals of $R$. Moreover, the members of any system $\text{S}\phantom{\rule{0.166667em}{0ex}}\subseteq \text{P}\phantom{\rule{0.166667em}{0ex}}\left(X\right)$ of finite character are in one-to-one correspondence with the conservative prime ideals contained in ${P}_{\text{S}}=\bigcup \{RS:S\in \text{S}\phantom{\rule{0.166667em}{0ex}}\}$, and the maximal members of $\text{S}\phantom{\rule{0.166667em}{0ex}}$ correspond to the maximal ideals contained in...

We study several choice principles for systems of finite character and prove their equivalence to the Prime Ideal Theorem in ZF set theory without Axiom of Choice, among them the Intersection Lemma (stating that if $\text{S}$ is a system of finite character then so is the system of all collections of finite subsets of $\bigcup \text{S}$ meeting a common member of $\text{S}$), the Finite Cutset Lemma (a finitary version of the Teichm“uller-Tukey Lemma), and various compactness theorems. Several implications between these statements...

It is known that for a nonempty topological space $X$ and a nonsingleton complete lattice $Y$ endowed with the Scott topology, the partially ordered set $[X,Y]$ of all continuous functions from $X$ into $Y$ is a continuous lattice if and only if both $Y$ and the open set lattice $\mathcal{O}X$ are continuous lattices. This result extends to certain classes of $\mathcal{Z}$-distributive lattices, where $\mathcal{Z}$ is a subset system replacing the system $\mathcal{D}$ of all directed subsets (for which the $\mathcal{D}$-distributive complete lattices are just the continuous...

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