Is this topological dimension or dimensionality as a manifold? Is \(\displaystyle S^p=\mathbb{S}^p=\left\{\bold{x}\in\mathbb{R}^{p-1}:\|\bold{x}\|=1\right\}\)?

Is this topological dimension or dimensionality as a manifold? Is \(\displaystyle S^p=\mathbb{S}^p=\left\{\bold{x}\in\mathbb{R}^{p-1}:\|\bold{x}\|=1\right\}\)?