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An $n-$sphere is a finite-dimensional arcwise-connected topological space.

Formal Definition

An $n-$sphere is a space that is homeomorphic to the $n-$dimensional sphere $S^n$ in $\mathbb{R}^{n+1}$, where $$S^n = \left \{ x \in \mathbb{R}^{n+1} : ||x|| = 1 \right \}, \quad \text{each } n = 1,2,...$$ An $n-$sphere is a continuum. A $1-$sphere is called a simple closed curve.[1]

See also


  1. Nadler, Sam B. Continuum Theory: An Introduction. Monographs and Textbooks in Pure and Applied Mathematics 158. New York: M. Dekker, 1992.