Does the definition of the set $S^n$ have implied algebraic structure? Let's say I wish to define a topological space $(S^1, \mathscr{O})$. I would imagine that such a space is only made of two structures, namely two sets.
However, the definition of $S^n$ is given by:
$$
S^1 = \left\{x\in \mathbb{R}^2 : x_1^2+x_2^2=1\right\}
$$
This set has an addition operator within its definition. Doesn't this mean that the definition of $S^2$ depends on the choice of addition in the field with $\mathbb{R}^2$ as its base set?
Wouldn't this further imply that the definition of this topological space depends not only on the two sets, but also on the choice of addition? 
 A: A topological space is a set in combination with a topology. Often, when the topology is clear from context, we just specify the set.
In this case, we define the set
$$ S^n = \bigg\{x\ \bigg|\sum_{i=0}^n x_i^2\bigg\} $$
as a subset of $\mathbb{R}^{n+1}$. Note that we are using the field structure of $\mathbb{R}$ to select the points of $S^n$; this implies nothing about any inherited algebraic structure in $S^n$. Indeed, if we use a different definition of "$+$" then the subset $S^n$ changes.$^*$ 
In this context, the topological space $S^n$ is defined as the set $S^n$ combined with the subspace topology $S^n$ inherits from $\mathbb{R}^{n+1}$.
$^*$ I applaud your attention to detail here. Mathematical progress often requires examining implicit assumptions and asking what can be understood when they are relaxed. So I would encourage you to consider: what are the different types of addition on $\mathbb{R}$? If there are any nontrivial examples, do any of these other "$+$" lead to spheres that are not homeomorphic to the standard $S^n$?
A: This particular definition of a sphere does indeed depend on the specific addition operation, and the result is the solution set of a polynomial equation, which is called an algebraic variety. The addition operation doesn't preserve the sphere, so it doesn't descend to an operation on the sphere, but it's still tied to the addition and multiplication operations since it is an algebraic variety.
If we only care about topology though, there are other subspaces homeomorphic to a sphere that are not algebraic varieties. 
