Given an infinite bounded subset $E$ of $\Bbb R$, is it always possible to choose an infinite subset $\{x_i\} \subset E$ and open neighborhoods $U_i$ of $x_i$ so that the $U_i \cap U_j = \varnothing$ if $i\neq j$?
Can we always select a discrete subset of an infinite subset of the line?
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real-analysis
1 Answers
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There is some $t_1$ so that both $\{x \in E \;: \; x < t_1\}$ and $\{x \in E\; : \; x > t_1\}$ are nonempty (e.g. you could take $t_1$ to be the midpoint of two distinct members of $E$. At least one of these is infinite: call that one $E_1 = E \cap (a_1, b_1)$. Take $x_1$ to be a point of the other one (which may or may not be infinite, but is nonempty), and $U_1$ an open neighbourhood of $x_1$ disjoint from $(a_1, b_1)$. Take $t_2 \in (a_1, b_1)$ so that $\{x \in E_1\;: \; x < t_2\}$ and $\{x \in E_1: x > t_2\}$ are nonempty, and continue the process.