0

Lets suppose that we have two sets, $S$ and $T$. We assume $S \subset R^n$ be bounded (and infinite) and likewise for $T$. Now suppose there is some correspondance relation $p(x)$ that maps subsets of $R^n$ to $R$.

Now, $T$ is composed in the following way: Take some element $s\in S$. Then create a set $U\subset R^n$ such that the supremum of the image of $p(U)$ is $p(s)$ and which does not contain $s$. Do this for every element of $S$, and take the union of all the sets created this way, and call that union $T$.

What I want to know is what properties does $p(\cdot)$ need to have in order for $\sup(p(S)) = \sup(p(T))$?

Antecedent
  • 45

1 Answers1

1

Rephrasing your question: You have some $S \subset R^n$ and a function $p : R^n \to R$. For each $s \in S$ you define some $U_s \subset R^n$ such that $s \notin U_s$ and $\sup p(U) = p(s)$. Then you define $T = \bigcup_{s \in S} U_s$.

Then $$\sup p(T)= \sup_{s \in S}\{\sup p(U_s)\} = \sup_{s \in S} p(s) \equiv \sup p(S).$$ I believe the first step (writing the supremum as a double supremum) is valid (e.g. see this).

angryavian
  • 89,882