The polynomial closure is not topological

We characterize the polynomial closure of a pseudo-convergent sequence in a valuation domain V of arbitrary rank, and then we use this result to show that the polynomial closure is never topological when V has rank at least 2.     

Every Cech-analytic Baire semitopological group is a topological group

Among other things, we prove the assertion given in the title. This solves a problem of Pfister.

     

Deligne's topological central extension is universal

We give a purely “local” proof of the fact that the topological central extension of G(k), G an absolutely almost simple algebraic group defined and isotropic over a nonarchimedean local field k, by the finite group μ(k) of roots of unity in k, constructed by Pierre Deligne, is a universal topological central extension of G(k).     

A homotopy equivalence that is not homotopic to a topological embedding

An open subset W of Sⁿ, n 6 or N = 4, and a homotopy equivalence ƒ: S² × S^{n − 4} → W are constructed having the property that ƒ is not homotopic to any topological embedding.     

The closed subalgebras of a topological algebra

Conclusion There are some interesting related problems which we have not solved One is to characterize the family of all closed subalgebras in the case where it is assumed that the algebraic operations are continuous in the topology. If the operations are not determined uniquely by the family of closed subalgebras, is there at least one determination in which they are continuous ? Another problem is to characterize the families of finitely generated closed subalgebras of a topological algebra.Supported by National Science Foundation Research Grant GP-3132.     

The dualizing spectrum of a topological group

To a topological group G, we assign a naive G-spectrum , called the dualizing spectrum of G. When the classifying space BG is finitely dominated, we show that detects Poincaré duality in the sense that BG is a Poincaré duality space if and only if is a homotopy finite spectrum. Secondly, we show that the dualizing spectrum behaves multiplicatively on certain topological group extensions. In proving these results we introduce a new tool: a norm map which is defined for any G and for any naive G-spectrum E. Applications of the dualizing spectrum come in two flavors: (i) applications in the theory of Poincaré duality spaces, and (ii) applications in the theory of group cohomology. On the Poincaré duality space side, we derive a homotopy theoretic solution to a problem posed by Wall which says that in a fibration sequence of fini the total space satisfies Poincaré duality if and only if the base and fiber do. The dualizing spectrum can also be used to give an entirely homotopy theoretic construction of the Spivak fibration of a finitely dominated Poincaré duality space. We also include a new proof of Browder's theorem that every finite H-space satisfies Poincaré duality. In connection with group cohomology, we show how to define a variant of Farrell-Tate cohomology for any topological or discrete group G, with coefficients in any naive equivariant cohomology theory E. When E is connective, and when G admits a subgroup H of finite index such that BH is finitely dominated, we show that this cohomology coincides with the ordinary cohomology of G with coefficients in E in degrees greater than the cohomological dimension of H. In an appendix, we identify the homotopy type of for certain kinds of groups. The class includes all compact Lie groups, torsion free arithmetic groups and Bieri-Eckmann duality groups. Received July 14, 1999 / Revised May 17, 2000 / Published online February 5, 2001     

The topological structure of a certain Menger space

In this paper we will reconsider the topological structure of Menger probabilistic normed spaces (briefly PN-spaces) under the t-norm M. We will prove that this topology is compatible with the topology induced by a countable and separating family of semi-norms, and hence the well-known theorems of classical functional analysis (such as the principle of uniform boundedness, open mapping and closed graph theorems) are valid in this context also. We will meanwhile obtain a method by which one may construct easily a large class of PN-spaces. Finally, using this method, we see that a certain subspace of bounded linear operators between PN-spaces, i.e. the class of strongly bounded linear operators, has a natural PN structure.     

非自治逆紧系统的拓扑压

主要研究了非自治逆紧系统上的拓扑压.给出了非自治逆紧系统上拓扑压的定义,得到了这种拓扑压关于集合Z的一些性质,并在同胚意义下,探讨了两个非自治逆紧系统上拓扑压的大小关系.     

非紧系统的次可加拓扑压

利用逆紧映射、容许覆盖等概念,将紧系统上的次可加拓扑压推广到非紧系统上,给出了次可加拓扑压的定义,并讨论了非紧系统上次可加拓扑压的性质.     

Every topological category is convenient for Gelfand duality

In this paper we generalize our work on Gelfand dualities in cartesian closed topological categories [42] to categories which are only monoidally closed. Using heavily enriched category theory we show that under very mild conditions on the base category function algebra functor and spectral space functor exist, forming a pair of adjoint functors and establishing a duality between function algebras and spectral spaces. Using recent results in connection with semitopological functors, we show that every (E,M)-topological category is endowed with at least oneconvenient monoidal structure admitting a generalized Gelfand duality. So it turns out that there is no need for a cartesian closed structure on a topological category in order to study generalized Gelfand-Naimark dualities.     

The topological behaviour of a diffeomorphism near a fixed point

A topological Linearization Theorem for flows by K. Palmer is extended to the case of maps.