## Introduction

Many admire the Euclidean space, and I am not an exception. I will try to catch the topological roundness of the $n$-ball in its greatest generality. I call the resulting axiomatized space to be a *razor*. If my definition turns out too general then (after getting interesting counter-examples) it will be a simple matter to narrow these definitions down (e.g. by finally agreeing--if necessary--to explicit homotopic constructions) till we are left with the most **possible general** characterizations (as opposed to simply **general**). Optimistically I hope for the minimal properties though (while among others the exotic cubes have to be considered).

**THE GOAL** (conjecture) is to show that for every razor $\ (X\ T)\ $ and $S\subseteq X$ (see below) there exists a non-negative integer $\ n\ $ and subspace $\ Y\subseteq X\ $ such that $\ X\setminus S\subseteq Y\ $ and pairs $\ (Y\ \,Y\!\cap\! S)\ $ and $\ (\mathbb I^n\ \,\partial(\mathbb I^n))\ $ are homeomorphic.

All topological spaces are assumed here to be $T_1$-spaces (each single-point set should be closed). Also by definition:

$$X^{(2)}\ :=\ X\times X\setminus \Delta_X$$

where $\ \Delta_X\ :=\ \{(x\ x) : x\in X\}$, for any set $\ X$.

At this time I am not making any assumptions about compactness, separability, finite dimension ... or even about the involved subsets being closed. But if you have to do them then do them, of course.

## Razors

A razor is a topological space $\ X\ $ for which there exists a razor structure. And a razor structure in $\ X\ $ is defined as a triple $\ (X\ S\ f)\ $ where $\ S\ $ is a subset of $\ X\ $, function $\ f: X^{(2)}\rightarrow\ S\ $ is continuous, ** f.p.p.** stands for

*the fixed point property*, and the following conditions hold:

- $\ X\ $ has the f.p.p.
- $\ S\ $ does not have the f.p.p.
- $\ \forall_{x\in X}\forall_{y\in S\setminus\{x\}}\ \ f(x\ y)\ =\ y$

**REMARK** Axiomat 1 is equivalent to property: $\ \ S\ $ is not a retract of $\ X\ \ $ (it's a standard observation in a situation like this).