On the variety of Euclidean point sets

TL;DR

This paper explores the topological diversity of subsets of the real line, constructing a continuum of non-homeomorphic compact sets and analyzing their properties in various topological and metric contexts.

Contribution

It introduces a continuum of non-homeomorphic compact subspaces of R without singleton components and examines how topological distinctions diminish under metric or order-theoretic perspectives.

Findings

Existence of a continuum of non-homeomorphic compact subsets of R.

Discrepancy between open sets and closed sets in the topological structure of R.

Classification of various topological and metrical types of subsets of R.

Abstract

We construct a continuum of non-homeomorphic compact subspaces of the real line R without singleton components. Thus from the purely topological point of view the real line contains not only more closed sets than open sets but also more closures of open sets than open sets. On the other hand, we show that this discrepancy vanishes either if the topological point of view is sharpened in the metrical or in the order-theoretical direction, or if R is replaced with R^n for dimension n>1. Furthermore, we track down a continuum of topological types of closed and totally disconnected subsets of R. In doing so we also track down a continuum of metrical types of infinite, discrete subsets of the unit interval [0,1]. (As a consequence, any countably infinite discrete space has a continuum of non-homeomorphic metrizable compactifications.)