#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... 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]]