General Sphere is Subspace of Closed Ball of Greater Dimension

Theorem

Let $S^{n - 1}$ be an $\paren {n - 1}$-sphere of the real Euclidean space $\R^n$.

Then $S^{n - 1}$ is a subspace of the closed Euclidean ball $\map { {B_1}^-} 0$ of $\R^n$.

Proof

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Examples

Dimension $1$

Consider the closed ball $\map { {B_1}^-} 0$ in the real Euclidean space $\R^2$.

The$1$-sphere $S^1 = \closedint {-1} 1$ is a subspace of $\map { {B_1}^-} 0$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ball
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ball