A Tate affinoid k-algebra is defined as a pair $(R,R^+)$ where $R$ is a Tate algebra and $R^+$ is an open and integrally closed subring of $R$ contained in the ring of powerbounded elements.


See for instance Defn 2.6 [here](http://www.math.uni-bonn.de/people/scholze/PerfectoidSpaces.pdf).


What is the (geometric) significance of requiring $R^+ $ to be integrally closed?


On a related note, what is the difference (geometrically) between total integral closure and integrally closed. It is surprisingly hard to find anything about total integral closures...



Significance of integrally closed in an affinoid algebra
The adic spectrum of $(R,R^+)$ is a set of continuous valuations on $R$ having norm $\leq 1$ on $R^+$. It is clear from the definition and basic properties of integral closures that relaxing the integral closedness of $R^+$ buys you nothing new: a valuation on $R$ has norm $\leq 1$ on a subring $R\_0 \subset R$ if and only if it has norm $\leq 1$ on the integral closure $R^+$ of $R\_0$ in $R$. I believe this is the main reason one often assumes $R^+$ is integrally closed. 


The difference between total integral closures and integral closures, roughly, corresponds to the difference between rank $1$ valuations and all valuations. But it might be better to ask a separate question about this as it is pure commutative algebra.


