Flimsy Spaces

TL;DR

This paper investigates the properties of n-flimsy spaces, proving their non-existence for n ≥ 3 and characterizing compact 2-flimsy spaces as dense, order-complete cyclically ordered sets, with extensions to path-connected variants.

Contribution

It establishes the non-existence of n-flimsy spaces for n ≥ 3 and characterizes compact 2-flimsy spaces, also exploring variants with path-connectedness.

Findings

No n-flimsy spaces exist for n ≥ 3.

Compact 2-flimsy spaces are dense, order-complete cyclically ordered sets.

Extensions to path-connected variants are examined.

Abstract

We study $n$-flimsy spaces, which are the topological spaces that remain connected when removing fewer than $n$ points but become disconnected when removing exactly $n$ points. We show that no such space exists for $n \geq 3$, and that the compact $2$-flimsy spaces are precisely the dense and order-complete cyclically ordered sets equipped with their order topology. Furthermore, we examine variants of the definition obtained by replacing connectedness by path-connectedness, where paths are either parametrized by $[0,1]$ or by arbitrary compact linear continua.