Name/Topological properties of the space of formal power series $\mathcal K [x]$ So,  a guest lecturer introduced a concept the other day in class.   Take a field $\mathcal K$ and then take the ring of formal power series on that ring, $\mathcal K[x]$.  Ignoring convergence in the normal sense, treating these only as formal power series,  he introduced a topology by (and here my memory is fuzzy)  taking the open balls as things which are in the ideals $<x>$, $<x^2>$, etc...
The gist of the point was that two formal power series were within a distance $k$ of each other if every coefficient from the $x^k$ on upwards to infinity agreed (So smaller $k's$ are actually 'closer'),  although he said this wasn't necessarilly a metric space, it did define a topology.
So,  my questions:   Does this topology have a name?  What, if any of the usual topological qualities does it have? (Connected, hausdorff, normal, seperated, what have you).
 A: Let $A$ be a commutative ring with unit. To define a ring topology on $A$ (means that the operations in $A$ are continuous) it suffices to define a system of neighbourhoods of $0$. (If you have this, by translation (continuous operation), you have a system of neighbourhoods of any point in $A$.) This data for all points of $A$ is equivalent to the data of a topology on $A$. (This is general : on any set $X$ giving yourself for each $x\in X$ a set of subsets of $X$ satisfying the axiom of a system of neighbourhood of $x$ is equivalent to the data of a topology on $X$.) Now, if $I$ is an ideal of $A$ you see that the set of subsets of $A$ containing $0$ and containing some power $I^n$ of $I$ fulfills the axioms to be a system of neighbourhoods of $0$. The topology defined in this way is called the $I$-adic topology. It is Hausdorff if and only if $\cap_{n\in \mathbf{N}} I^n = (0)$. Hausdorff is another name for separated. If $A$ is a discrete valuation ring, the topology is metrizable. (See the case of $A = k[X]$ below for the definition of the distance.)
In the case of $ A = k[X]$ and $I = (x)$ you get the so called $(x)$-adic topology. This topology is separated, as no non zero polynomial could be divided by abritrary large powers of $X$. This topology is metrizable : the distance between $P$ and $Q$ is $0$ if $P = Q$, and $e^{-d}$ where $d$ is the biggest integer such that $X^d$ divides $P-Q$ if the latter is non zero. (Metrizability also implies separability.) This distance is ultrametric (also called non-archimedean), so that two open balls are either disjoints or one of them contains the other. All triangles are isocele. The topology is totally disconnected (each singleton is its own connected component). This metric space is not complete, it's completion being the ring of formal series $k[[X]]$, which will be compact if the residue field $k$ is finite.
Ultimate (IMO) reference for this : Bourbaki, Algèbre Commutative, Chapter III, paragraphs 2 and 3.
