Is a torus a subset of $\mathbb{R}^3$ or $\mathbb{R}^4$? Defining a torus $T$ as $S^1 \times S^1$, it should follow that $T \subseteq \mathbb{R}^4$.   But you can also think of a torus as a bagel, which means it's a subset of $\mathbb{R}^3$.  Can anyone clarify this point?
 A: Neither!  A torus is an abstract topological space defined, as you say, to be the product $S^1 \times S^1$.  Since, as it happens, $S^1 \subset \mathbb{R}^2$, we get a natural embedding of "the" torus into $\mathbb{R}^4$.  However, there are of course other ways of representing this space; one can change coordinates, apply functions to it, and so on.  And among these alterations are some that "project" the torus down into $\mathbb{R}^3$, where it resembles a bagel.
This is a little bit of an aberration, since not every two-dimensional manifold can be embedded in $\mathbb{R}^3$, the Klein bottle being a counterexample (also, this is related to your question in that it is more or less a twisted torus).  And more generally, an $n$-dimensional manifold can be embedded in $\mathbb{R}^{2n}$ but not necessarily lower.  So if you insist on thinking of a manifold as "being" a subset of some $\mathbb{R}^N$, that's the one to choose.
A: Neither. The torus is generally defined as an abstract topological space, independently of any "embedding" into another space. So in the strict sense, it is incorrect to say $T\subseteq \mathbb R^4$ or $T\subseteq \mathbb R^3$.
However, we often talk about abstract spaces like the torus as subsets of other spaces, commonly $\mathbb R^n$ for some $n$. How do we resolve this? The answer is something called an embedding. We say that a space $A$ can be embedded into a space $B$ if there is a continuous function $f:A\to B$ which maps $f$ injectively onto a subset of $B$, such that $f$ restricted to $B$ has a continuous inverse in the subspace topology$^1$. What you're noticing is that there are embeddings of the torus into $\mathbb R^3$ and $\mathbb R^4$.



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*Note that this is for topological spaces. If we add additional structure to the space, then the embedding is required to preserve this structure in a suitable way. It is common for example to give the torus a manifold structure, and the embedding is then required to be an embedding of manifolds, which is somewhat involved to define. Depending on the additional structure imposed, the torus may fail to embedd into $\mathbb R^3$, as Matt points out.

A: These are two different embeddings of the same topological space, i.e. the surface of the bagel in $\mathbb R^3$ and the product of two circles in $\mathbb R^4$ are homeomorphic. Note that they are not isometric though; the torus embeded in $\mathbb R^4$ as a product of circles is flat, while the surface of the bagel has regions of positive curvature and of negative curvature.
