We show that two analytic function germs (C, 0) → (C, 0) are topologically right equivalent if and only if there is a one-to-one correspondence between the irreducible components of their zero sets, that preserves the multiplicites of these components, their Puiseux pairs, and the intersection numbers of any pairs of distinct components. By Zariski [7] and Burau [1], the topological type of an ...

0965 Contents 1. Introduction 1 2. Some topology 2 3. Local isomorphisms 4 4. Ind-Zariski algebra 6 5. Constructing w-local affine schemes 6 6. Identifying local rings versus ind-Zariski 10 7. Ind-´ etale algebra 14 8. Constructing ind-´ etale algebras 15 9. Weaklyétale versus pro-´ etale 18 10. Constructing w-contractible covers 18 11. The pro-´ etale site 21 12. Points of the pro-´ etale site...

We consider a closed semi-algebraic set X ⊂ R and a C semi-algebraic function f : R → R such that f|X has a finite number of critical points. We relate the topology of X to the topology of the sets {f ∗ α}, where ∗ ∈ {≤,=,≥} and α ∈ R, and the indices of the critical points of f|X and −f|X . We also relate the topology of X to the topology of the links at infinity of the sets {f ∗ α} and the in...

Based on a recent work of Thomas Bauer’s [1] reproving the existence of Zariski decompositions for surfaces, we construct a b-divisorial analogue of Zariski decomposition in all dimensions.

It is shown that the ideal theories of the fields of all meromorphic functions on any two noncompact Riemann surfaces are isomorphic. Further, various new representation and factorization theorems are proved. Introduction. Throughout this paper let X and Y denote noncompact (connected) Riemann surfaces. Let A(X) (or A for short), denote the ring of all analytic functions on X, and let F(X) (or ...

Let C A n be a geometrically irreducible aane curve of (geometric) genus 0 deened over Z such that C(Z) is an innnite set. We use elementary methods to describe the distribution of C(Z) in C(R) relative to the real topology. The study of integral points on aane curves has a long history. The deenitive niteness theorem, due to Siegel 5], says that a geometrically irreducible aane curve C has onl...

Following the categorical approach to universal algebra through algebraic theories, proposed by F.~W.~Lawvere in his PhD thesis, this paper aims at introducing a similar setting for general topology. The cornerstone of the new framework is the notion of emph{categorically-algebraic} (emph{catalg}) emph{topological theory}, whose models induce a category of topological structures. We introduce t...