Why torus space we could see it in $\mathbb R^3$

Question

Anyone know why Torus we can see it in $\mathbb{R}^3$ ? I don't understand why torus as homeomorphic to $S^1 \times S^1$ and see it in $\mathbb{R}^3$, if $ S^1 \times S^1 \subset \mathbb{R}^4$

Answer 1

You can see $S^1$ as a subspace of $\mathbb{R}^2$, therefore, you can see $S^1\times S^1$ as a subspace of $\mathbb{R}^4$. However, this doesn't mean it cannot be homeomorphic to a subspace of an euclidean space of lower dimension. Take for example the topological product of two straight lines ( both of which you can see as subspaces of $\mathbb{R}^2$). The product is homeomorphic to $\mathbb{R}^2$ itself, so you don't need four, or even three dimensions.