Consider the following topologies on $\mathbb{R}$:
$\mathscr{T}_2$: the finite complement topology.
$\mathscr{T}_3$: the lower limit topology, having all sets $[a,b)$ as basis.
$\mathscr{T}_4$: the upper limit topology, having all sets $(a,b]$ as
basis.$\mathscr{T}_5$: the topology having all sets $(-\infty, a)$ as basis.
Describe for each of them which sequences converge to which limits.
I'm a little stumped by what this question is asking. There are infinite sequences in $\mathbb{R}$ that converge to different limits. There are infinite sequences that don't converge. I don't see what the question is even asking.
I suspect this problem may have something to do with Hausdorff properties? $\mathscr{T}_2$ and $\mathscr{T}_5$ are not Hausdorff topologies, while $\mathscr{T}_3$ and $\mathscr{T}_4$ are Hausdorff. With a Hausdorff topology, a sequence of points converges to at most one point. With non-Hausdorff topologies, that isn't necessarily true.
Best Answer
It is easy to check the following.
A sequence $\{a_n\}$ of points of $\Bbb R$ converges to a limit $a\in\Bbb R$ iff
$\mathscr{T}_2$: $\{a_n\}$ attains each value distinct from $a$ only finitely many times.
$\mathscr{T}_3$: All by finitely many $a_n$ are not smaller than $a$ and for each $\varepsilon>0$, all by finitely many $a_n$ are not bigger than $a+\varepsilon$.
$\mathscr{T}_4$: All by finitely many $a_n$ are not bigger than $a$ and for each $\varepsilon>0$, all by finitely many $a_n$ are not smaller than $a-\varepsilon$.
$\mathscr{T}_5$: For each $\varepsilon>0$, all by finitely many $a_n$ are not bigger than $a+\varepsilon$.