Letg:U→ℝ (U open in ℝn) be an analytic and K-subanalytic (i. e. definable in ℝanK, whereK, the field of exponents, is any subfield ofℝ) function. Then the set of points, denoted Σ, whereg does not admit an analytic extension is K-subanalytic andg can be extended analytically to a neighbourhood of Ū.
We prove rectilinearization and uniformization theorems for K-subanalytic (∝anK-definable) sets and functions using the Lion-Rolin formula. Parallel reasoning gives standard results for the subanalytic case.
We give a topological version of a Bertini type theorem due to Abhyankar. A new definition of a branched covering is given. If the restriction of the natural projection π: Y × Z → Y to a closed set V ⊂ Y × Z is a branched covering then, under certain assumptions, we can obtain generators of the fundamental group π₁((Y×Z).
We prove that the semialgebraic, algebraic, and algebraic nonsingular points of a definable set in o-minimal structure with analytic cell decomposition are definable. Moreover, the operation of taking semialgebraic points is idempotent and the degree of complexity of semialgebraic points is bounded.
We begin a systematic study of the category GTS of generalized topological spaces (in the sense of H. Delfs and M. Knebusch) and their strictly continuous mappings. We reformulate the axioms. Generalized topology is found to be connected with the concept of a bornological universe. Both GTS and its full subcategory SS of small spaces are topological categories. The second part of this paper will also appear in this journal.
This is the second part of A. Piękosz [Ann. Polon. Math. 107 (2013), 217-241]. The categories GTS(M), with M a non-empty set, are shown to be topological. Several related categories are proved to be finitely complete. Locally small and nice weakly small spaces can be described using certain sublattices of power sets. Some important elements of the theory of locally definable and weakly definable spaces are reconstructed in a wide context of structures with topologies.
A correct formulation of the Lion-Rolin Preparation Theorem for logarithmic-subanalytic functions (LA-functions) is given.
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