Thread: *|*Interesting Limit Set*|* View Single Post
 Emeritus Sci Advisor PF Gold P: 16,091 The only other thing I recall seeing is just a mild generalization: instead of having the sets actually be subsets of one another, you have a chain of maps: $$S_0 \rightarrow S_1 \rightarrow S_2 \rightarrow S_3 \rightarrow \cdots$$ and then you can take the limit of this diagram: $$\begin{tabular}{ccccccccc} S_0 & \rightarrow & S_1 & \rightarrow & S_2 & \rightarrow & S_3 & \rightarrow & \cdots \\ \downarrow & & \downarrow & & \downarrow & & \downarrow & &\cdots \\ S & = & S & = & S & = & S & = & \cdots \end{tabular}$$ Of course, this limit won't be unique, since I could replace S with anything of the same cardinality, and just compose all of those downward arrows with a corresponding bijection. (If all of the rightward arrows are identity maps, then you get something isomorphic to the ordinary set limit)