On the limit closure of a sequence of elements in local rings

We present a systematic study for the limit closure $(\underset{‾}{x}{)}^{\lim }$ of a sequence of elements $\underset{‾}{x}$ (eg. a system of of parameters) in a local ring. Firstly, we answer the question which elements are always contained in the limit closure of a system of parameters. Then we apply this result to give a characterization of systems of parameters which is a generalization of previous results of Dutta and Roberts in \cite{DR} and of Fouli and Huneke in \cite{FH}. We also prove a topological characterization of unmixed local rings. In two dimensional case, we compute explicitly the limit closure of a system of parameters. Some interesting examples are given.