Hurewicz tests: separating and reducing analytic sets in a concise way

نویسنده

  • Tamás Mátrai
چکیده

If we add the well-known fact that any subset N of the Cantor set C homeomorphic to the irrationals is never Fσ, we have given all the reasons why the pair (C, N) is called the Hurewicz test for Fσ sets. Since the work of Hurewicz Theorem 1 has been strengthened in many successive steps. We will discuss this in the next section. Before doing this we recall some basic definitions in descriptive set theory. Our basic reference for this is [1]. All the spaces we consider are Polish. The product spaces 2 and ω are endowed with their usual product topology, that we denote by τ2ω and τωω respectively. For two arbitrary sets A, B in a topological space X we say that A is Wadge reducible to B, which we denote as A ≤W B, if there is a continuous function f : X → X such that f−1(B) = A. This preordering gives rise to the equivalence relation A ≡W B ⇐⇒ A ≤W B ∧ B ≤W A. If

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Separation of analytic sets by rectangles of low complexity

We provide Hurewicz tests for the separation of disjoint analytic sets by rectangles of the form Γ× Γ for Γ,Γ ∈ {Σ01,Π 0 1,Π 0 2}.

متن کامل

Hurewicz-like Tests for Borel Subsets of the Plane

Let ξ ≥ 1 be a countable ordinal. We study the Borel subsets of the plane that can be made Πξ by refining the Polish topology on the real line. These sets are called potentially Πξ . We give a Hurewicz-like test to recognize potentially Πξ sets. 1. Preliminaries in dimension one Let us recall some results in dimension one before studying Borel subsets of the plane. In descriptive set theory, a ...

متن کامل

A Functional Characterization of the Hurewicz Property

For a Tychonoff space $X$, we denote by $C_p(X)$ the space of all real-valued continuous functions on $X$ with the topology of pointwise convergence.  We study a functional characterization of the covering property of Hurewicz.

متن کامل

The Hurewicz Theorem

The fundamental group and homology groups both give extremely useful information, particularly about path-connected spaces. Both can be considered as functors, so we can use these constructional invariants as convenient guides to classifying spaces. However, though homology groups are often easy to compute, the fundamental group sometimes is not. In fact, it is often not even obvious when a spa...

متن کامل

Accuracy Assessment of Interphase Fluorescence In-Situ Hybridization on Uncultured Amniotic Fluid Cells

a:4:{s:10:"Background";s:431:"Parental anxiety while waiting for the results of amniocentesis has been investigated by many authors. It seems that the implementation of faster techniques such as fluorescence in-situ hybridization (FISH) will have some benefits in reducing this anxiety. Besides the patients' attitudes to choosing this method, gynecologists who are the persons responsible for tre...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2004