One-Sided Notion of Topological Closure

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.

Answers 1

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.

José Carlos Santos
José Carlos Santos
August 06, 2018 23:15 PM

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