Microhomogeneity

Abstract

A space X is microhomogeneous if for every $p,q\in X$ there is a homeomorphism h from a neighborhood of p onto a neighborhood of q such that $h(p)=q$. We show that many cardinal bounds obtained using homogeneity or power homogeneity can be obtained using microhomogeneity. Since microhomogeneous spaces need not be either homogeneous or power homogeneous, this extends those cardinal bounds to a broader class of spaces. We also explore possible connections between microhomogeneity and both homogeneity and power homogeneity, providing a simple example of a connected microhomogeneous space that is not homogeneous. We also give examples of microhomogeneous spaces in which no open set is homogeneous, and a microhomogeneous space which is not uniformly microhomogeneous in the sense that for every $p\in X$ and every neighborhood U of p there is $q\in X$ which does not have a neighborhood that is homeomorphic to U. In the process of establishing the properties of the last example we obtain the following result, which is of independent interest. Let X be a locally compact, homogeneous, and strongly locally homogeneous space (such as a manifold). Let A and B be countable dense subsets of X and let $\{A_i:i\in I\}$ and $\{B_i:i\in I\}$ be partitions of A and B respectively into dense subsets of X. For every $a_0,b_0\in X$ there is a homeomorphism $F:X\to X$ such that $F(a_0)=b_0$ and $F[A_i-\{a_0\}]=B_i-\{b_0\}$.

Introduction

The focus of this paper is on homogeneous spaces and their generalizations, in particular, local versions of homogeneity. A topological space X is homogeneous if for each $x,y\in X$ there is an auto-homeomorphism $f:X\to X$ with $f(x)=y$. It is power homogeneous if there is a cardinal κ such that $X^{\kappa }$ is homogeneous. Homogeneous spaces include Euclidean space, topological groups, and connected manifolds. A space is homogeneous if, roughly, the space looks the same at each point. This informal description hints at a local version of homogeneity. Consider the space that is the disjoint union of $\mathbb{R}$ and ${\mathbb{S}}_1$. The space is not homogeneous since we cannot swap points on the line and the circle. Nevertheless zooming into any point we see a copy of $\mathbb{R}$. This motivates the definition of microhomogeneity

One version of local homogeneity is microhomogeneity, introduced by van Dantzig in [9]. A space X is microhomogeneous if and only if for every $x,y\in X$ there are neighborhoods U and V of x and y respectively and a homeomorphism $f:U\to V$ such that $f(x)=y$.

In general, the cardinality of a topological space can be large relative to topological measures such as tightness, cellularity, or weight. For example, it is easy to find spaces with $|X|=2^{2^{w(X)}}$. However, for homogeneous spaces, the cardinality is more controlled. The study of cardinality bounds in homogeneous spaces begins with results by van Douwen [10] and Arhangel'skiĭ [2] and continues to present day with work by Carlson, Ridderbos, Gotchev, Bella, de la Vega, and others.

The results of the paper are organized into Sections 3 and 4. We are interested not only in generalizing results about homogeneous spaces, but exploring which techniques extend to the microhomogeneous context and which do not. So, for example, in Section 3, we present two proofs of van Douwen's theorem (Proof of 3.1 and Proof of 3.2), because these proofs extend two distinct techniques. We also generalize numerous cardinality bounds using one-to-one functions in 3.2. In Section 4, we present various examples of microhomogeneous spaces that fail to be homogeneous in increasingly severe ways.