在Dedekind、单调或Scott不完全序域中使用网和可定义性问题
Using nets in Dedekind, monotone, or Scott incomplete ordered fields and definability issues
摘要：
给定一个Dedekind不完全序域，利用一对分别递增或递减到同一点的收敛间隙网，得到了序域的Dedekind完全性的进一步等价判据：定义在有界闭区间上的连续一一函数将该区间的内部映射到象的内部.其次，证明了在任何单调不完全序域的闭有界区间上，存在连续的非一致连续的无界函数，其值域不是闭的，并且存在连续的1-1函数，其每个内点映射到（映象的）一个内点，但不是开的.这些都是通过适当的网络来实现的，在间隙中共尾或在其互补中共起始。在我们的第三个主要定理中，一个序域被构造，它具有参数可定义的正则间隙，但没有$\emptyset$ -可定义的发散柯西函数（而我们表明，在参数是或不允许的两种情况下，任何可定义的发散柯西函数产生一个可定义的正则间隙）。我们的证明所提到的独立性结果使用的存在性的子环中的广义幂级数序域的有理指数和实系数组成的系列没有无穷小的条款，最近建立的D。皮特鲁
英文摘要：
Given a Dedekind incomplete ordered field, a pair of convergent nets of gaps which are respectively increasing or decreasing to the same point is used to obtain a further equivalent criterion for Dedekind completeness of ordered fields: Every continuous one-to-one function defined on a closed bounded interval maps interior of that interval to the interior of the image. Next, it is shown that over all closed bounded intervals in any monotone incomplete ordered field, there are continuous not uniformly continuous unbounded functions whose ranges are not closed, and continuous 1-1 functions which map every interior point to an interior point (of the image) but are not open. These are achieved using appropriate nets cofinal in gaps or coinitial in their complements. In our third main theorem, an ordered field is constructed which has parametrically definable regular gaps but no $\emptyset$-definable divergent Cauchy functions (while we show that, in either of the two cases where parameters are or are not allowed, any definable divergent Cauchy function gives rise to a definable regular gap). Our proof for the mentioned independence result uses existence of infinite primes in the subring of the ordered field of generalized power series with rational exponents and real coefficients consisting of series with no infinitesimal terms, as recently established by D. Pitteloud.
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