Has the following notion already been studied? I am really stuck on a problem that seems to require some general results concerning this:
Let $A$ be a subspace of $\mathbb{R}$. Define an operation called 'left closure' to be $lcl(A)$, that sends $A$ to the subset
$\lbrace x \in \mathbb{R} : \text{ there exist } y_n \in A \text{ s.t. } y_n \leq x$ $\forall$ $n \text{ and } y_n \rightarrow x \rbrace$
So this is like the left-sided closure of $A$; it contains all the points that $A$ 'converges to' from the left. We can define $rcl(A)$ similarly. I am wondering what the best way to think about these objects is. There seem like too many choices. In my particular situation, I am dealing with topologically unpleasant objects (uncountable unions of pw-disjoint Cantor Sets), so anything that is well suited to such applications is preferable.
To ask more of a question than a reference request, what can be said about infinite unions of Cantor sets, either countably or uncountable? Anything that falls out of the definition?
The problem that I am especially interested in is this: If $C$ is the union of an uncountable collection of pw disjoint Cantor Sets $C_\alpha$, is it possible to apply an order to the indices taking values from $[0,1]$ so that the union of $C_\alpha, 0 \leq \alpha \leq \alpha_0$ is right-closed in $C$ for any fixed $\alpha_0$? That is to say that any collection of sets 'to the left' of some $C_{\alpha_0}$ doesn't accumulate to one of its points from the right.
Thanks! I tagged some topics that seem most likely to contribute, but if anyone has other suggestions, then that would also be helpful.