Example 3.1.1.1.3 (009D): The Zero Sphere

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Table of Contents
Part 1: Sets
Chapter 3: Pointed Sets
Section 3.1: Pointed Sets
Subsection 3.1.1: Foundations
Example 3.1.1.1.3: The Zero Sphere (cite)

Statistics Cite

Example 3.1.1.1.3 ▶ The Zero Sphere

The $0$-sphere^[1] is the pointed set $\smash {\webleft (S^{0},0\webright )}$^[2] consisting of:

The Underlying Set. The set $S^{0}$ defined by

\[ S^{0} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ 0,1\webright\} . \]
The Basepoint. The element $0$ of $S^{0}$.

Footnotes

[1] Further Terminology: In the context of monoids with zero as models for $\mathbb {F}_{1}$-algebras, the $0$-sphere is viewed as the underlying pointed set of the field with one element.

[2] Further Notation: In the context of monoids with zero as models for $\mathbb {F}_{1}$-algebras, $S^{0}$ is also denoted $\webleft (\mathbb {F}_{1},0\webright )$.

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