Spaces with function definition in G. Kempf, Algebraic Varieties

Question

I have started reading G. Kempf's Algebraic Varieties, and on first page they define a space with functions to be a topological space with a $k$-algebra of $k$-valued regular functions assigned to each open set. Then they define a regular function $f$ to be locally regular and that $1/f$ be regular on the open set $D(f) = \{x \in X | f(x) \neq 0\} $. My question is why does it mean that the sum of two regular functions will also be regular, for example?

Answer 1

Just so this question has an answer, I copy my comment.

By definition the set of regular functions over a fixed open subset is equipped with a $\Bbbk$-algebra structure, so their sum defines a regular function.

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An an aside, I think a modern sheaf-theoretic perspective on "spaces with functions" might be helpful. The structure of a 'space with $\Bbbk$-valued regular functions' on a topological space $X$ is precisely a subsheaf $\mathcal O_X$ of the sheaf of set-functions $f:X\to \Bbbk$ which has the following pair of properties. First, $f\in \mathcal O_X(U)\implies D_f\subset U$ is open where $D_f$ is the set of point on which $f$ is nonzero. Second, $f\in \mathcal O_X(U)\implies \frac 1f\in \mathcal O_X (D_f)$.

In my view, this characterization is more convenient for developing the theory of varieties, and also for relating it to the theory of schemes.