A fine property of the non-empty countable dense-in-self set in the real line

A fine property of the
non-empty countable dense-in-self set in the real line

Zujin
Zhang

School
of Mathematics and Computer Science,

GannanNormalUniversity

Ganzhou
341000, P.R. China

[email protected]

MSC2010: 26A03.

Keywords: Dense-in-self set;
countable set.

Abstract:

Let E?R1

be non-empty, countable, dense-in-self, then we shall show that Eˉ?E

is dense in Eˉ

.

Introduction and the main
result

As is well-known, Q?R1

is countable, dense-in-self (that is, Q?Q′=R1

); and R1?Q

is dense in R1

.

We generalize this fact as

Theorem 1.
Let E?R1

be non-empty, countable, dense-in-self, then Eˉ?E

is dense in Eˉ

.

Before proving Theorem 1, let us recall
several related definitions and facts.

Definition 2. A set E

is closed iff E′?E

. A set E

is dense-in-self iff E?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 R1

.

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

E=???n≥1(an,bn)??c,

where (ai,bi)

, (aj,bj)

(i≠j

) have no common points.

Proof of Theorem 1

Since E

is dense-in-self, we have E?E′

, Eˉ=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′?E≠?

.

Now that E′

is complete, we see by Lemma 4,

E′c=?n≥1(an,bn).

For ? x∈E′

, ? δ>0

, we have

[x?δ,x+δ]∩E′=([x?δ,x+δ]∩(E′?E))∪([x?δ,x+δ]∩E).(1)

By analyzing the complement of [x?δ,x+δ]∩(E′?E)

, we see [x?δ,x+δ]∩E′

(minus {x?δ}

if x?δ

equals some an

, and minus {x+δ}

if x+δ

equals some bn

) is compelete, thus has power c

. Due to the fact that E

is countable, we deduce from (1)

that

[x?δ,x+δ]∩(E′?E)≠?.

This completes the proof of Theorem
1.

A fine property of the non-empty countable dense-in-self set in
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A fine property of the non-empty countable dense-in-self set in
the real line