Characterization of a subset of [0,1] $II$ 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!
 A: 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.
