by John Samples
Last Updated August 06, 2018 23:20 PM

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?

Thanks! I tagged some topics that seem most likely to contribute, but if anyone has other suggestions, then that would also be helpful.

What you are talking about is the closure with respect to the upper limit topology in $\mathbb R$, that is, the topology for which the intervals $(a,b]$ form a basis.

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