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My question follows the previous one

Characterization of a subset of $[0,1]$

But I don't know whether it is correct to ask again with a new title.

Thanks a lot for pointing the mistake and I should reformulate my question.

Let $T\subset [0,1]$ be a subset satisfying the following property:

For every $t\in T\backslash\{1\}$ and any countable subset $D\subset [0,1]$, there exists a decreasing sequence $(t_n)_{n\ge 1}\subset T\backslash D$ such that

$$\lim_{n\to\infty}t_n=t$$

Obviously, if $T$ is $T=[a,b)\subset [0,1]$ satisfy the previous property. Now I would like to obtain a characterization of such $T$, does someone have an idea? Thx for your reply!

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    $\begingroup$ So, for example, you cannot have $t \in T$ but an interval $(t,t+\delta)$ disjoint from $T$, since there is no way for $t_n>t$ to converge to $t$. But perhaps that is the only restriction. $\endgroup$
    – Gerald Edgar
    Commented Sep 22, 2014 at 12:16
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    $\begingroup$ Given the possibility of excluding arbitrary countable subsets, I guess the restriction has to be that for $t\in T, t\neq 1$, the intersection $T\cap[t,t+\delta]$ must be uncountable for all $\delta>0$. $\endgroup$
    – Klaus Draeger
    Commented Sep 22, 2014 at 13:06

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To relate your property with a well-known notion in topology, let denote $\rho$ the Right Half-Open topology on $\mathbb{R}$, that is, the topology generated by the family of all right half-open intervals $[a,b)$. It turns out to be non-metrizable, yet first countable, so that topological notions have a sequential characterization. Also recall that a condensation point $t$ of a topological space $T$ is a point all of whose nbds are uncountable.

We can therefore rephrase your property, for $T\subset [0,1]$,

For every $t\in T\backslash\{1\}$ and any countable subset $D\subset[0,1]$, there exists a decreasing sequence $(t_n)_{n\ge 1}\subset T\backslash D$ such that $$\lim_{n\to\infty}t_n=t $$

saying equivalently that each of its points $t<1$ is a condensation point in the topology induced by $\mathbb{\rho}$, that is (as observed in comment by Klaus Draeger) every $\rho$-nbd of $t$ meets uncountably many elements of $T$.

For various properties of the right half-open topology (aka lower limit topology and Sorgenfrey topology) you may like to check Steen & Seebach's Counterexamples in topology.

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  • $\begingroup$ Thx so much to Pietro Majer for his reply! $\endgroup$
    – CodeGolf
    Commented Sep 23, 2014 at 9:34
  • $\begingroup$ Now for the previous $T$, if we denote by $E\subset T$ its subset of all condensation points, what is the behavior of $T\backslash E$? $\endgroup$
    – CodeGolf
    Commented Sep 23, 2014 at 10:32
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    $\begingroup$ Is it $T\backslash E$ is at most countable? $\endgroup$
    – CodeGolf
    Commented Sep 23, 2014 at 12:05
  • $\begingroup$ I'd think so. Fact: a Lindelöf topological space X that has no condensation points is countable (easy form definitions). Also, every subspace of the Sorgenfrey line is Lindelöf. And X:=T\E has no condensation points. $\endgroup$
    – Pietro Majer
    Commented Sep 23, 2014 at 14:02
  • $\begingroup$ We can show that $E$ is closed and thus $T\backslash E$ is open, but an open subset of a Lindel\"of space may not be Lindel\"of. Moreover, is the Cantor set a counterexample? $\endgroup$
    – CodeGolf
    Commented Sep 23, 2014 at 21:15

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