A fine property of the non-empty countable dense-in-self set in the real line
Zujin Zhang
School of Mathematics and Computer Science,
Gannan Normal University
Ganzhou 341000, P.R. China
MSC2010: 26A03.
Keywords: Dense-in-self set; countable set.
Abstract:
Let $Esubset bR^1$ be non-empty, countable, dense-in-self, then we shall show that $ar Es E$ is dense in $ar E$.
1. Introduction and the main result
As is well-known, $bQsubsetbR^1$ is countable, dense-in-self (that is, $bQsubset bQ'=bR^1$); and $bR^1s bQ$ is dense in $bR^1$.
We generalize this fact as
Theorem 1. Let $Esubset bR^1$ be non-empty, countable, dense-in-self, then $ar Es E$ is dense in $ar E$.
Before proving Theorem 1, let us recall several related definitions and facts.
Definition 2. A set $E$ is closed iff $E'subset E$. A set $E$ is dense-in-self iff $Esubset E'$; that is, $E$ has no isolated points. A set $E$ is complete iff $E'=E$.
A well-known complete set is the Cantor set. Moreover, we have
Lemma 3 ([I.P. Natanson, Theory of functions of a real variable, Rivsed Edition, Translated by L.F. Boron, E. Hewitt, Vol. 1, Frederick Ungar Publishing Co., New York, 1961] P 51, Theorem 1). A non-empty complete set $E$ has power $c$; that is, there is a bijection between $E$ and $bR^1$.
Lemma 4 ([I.P. Natanson, Theory of functions of a real variable, Rivsed Edition, Translated by L.F. Boron, E. Hewitt, Vol. 1, Frederick Ungar Publishing Co., New York, 1961] P 49, Theorem 7). A complete set $E$ has the form
$$ex E=sex{igcup_{ngeq 1}(a_n,b_n)}^c, eex$$
where $(a_i,b_i)$, $(a_j,b_j)$ ($i eq j$) have no common points.
2. Proof of Theorem 1。
Since $E$ is dense-in-self, we have $Esubset E'$, $ar E=E'$. Also, by the fact that $E''=E'$, we see $E'$ is complete, and has power $c$. Note that $E$ is countable, we deduce $E's E eq vno$.
Now that $E'$ is complete, we see by Lemma 4,
$$ex E'^c=igcup_{ngeq 1}(a_n,b_n). eex$$
For $forall xin E'$, $forall delta>0$, we have
$$eelabel{dec} [x-delta,x+delta]cap E'=sex{[x-delta,x+delta]cap (E's E)} cupsex{[x-delta,x+delta]cap E}. eee$$
By analyzing the complement of $[x-delta,x+delta]cap (E's E)$, we see $[x-delta,x+delta]cap E'$ (minus $sed{x-delta}$ if $x-delta$ equals some $a_n$, and minus $sed{x+delta}$ if $x+delta$ equals some $b_n$) is compelete, thus has power $c$. Due to the fact that $E$ is countable, we deduce from eqref{dec} that
$$ex [x-delta,x+delta]cap (E's E) eq vno. eex$$
This completes the proof of Theorem 1.