#Contents# * table of contents {:toc} ## Idea ## Limits in the real numbers satisfy certain algebraic properties, while the usual notion of convergence space, [[Hausdorff space]], and topological space are purely analytic. ## Definition ## Let $R$ be a [[ring]] and a [[Hausdorff space]]. $R$ is a Hausdorff ring if the ring operations are pointwise continuous... This is dumb, that the ring operations are pointwise continuous follows from the ring axioms and fact that product Hausdorff spaces exist. Let $I$ be a [[directed type]] and $x:I \to R$ and $y:I \to R$ be nets which converges to $a:R$ and $b:R$ respectively. By the definition of a function algebra, one could define pointwise the nets $0$, $x + y$, $1$, $-x$, $x \cdot y$ for natural number $n$. Let us denote the limit of a net as the partial function $$\lim_{i:I, i \to \infty} (-)(i): (I \to R) \to R$$ $R$ is a __Hausdorff ring__ if the limit of a net partial function preserves the ring operations, provided the limit exists: $$p_{0}:\lim_{i:I, i \to \infty} 0(i) = 0$$ $$x: I\to R, y:I \to R, \lim_{i:I, i \to \infty} x(i):R, \lim_{i:I, i \to \infty} y(i):R \vdash p_{+}(x, y):\lim_{i:I, i \to \infty} (x + y)(i) = \lim_{i:I, i \to \infty} x(i) + \lim_{i:I, i \to \infty} y(i)$$ $$x: I\to R, \lim_{i:I, i \to \infty} x(i):R \vdash p_{-}(x):\lim_{i:I, i \to \infty} (-x)(i) = -\lim_{i:I, i \to \infty} x(i)$$ $$p_{1}:\lim_{i:I, i \to \infty} 1(i) = 1$$ $$x: I\to R, y:I \to R, \lim_{i:I, i \to \infty} x(i):R, \lim_{i:I, i \to \infty} y(i):R \vdash p_{\cdot}(x, y):\lim_{i:I, i \to \infty} (x \cdot y)(i) = \lim_{i:I, i \to \infty} x(i) \cdot \lim_{i:I, i \to \infty} y(i)$$ ## See also ## * [[ring]] * [[Hausdorff space]]