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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.

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  • $\begingroup$ I would guess the question aks for characterizations of what convergence in each of the topologies means.For $T_3, T_4$ the answer is probably something along the lines of the 4th property on wiki. $\endgroup$ Feb 13, 2020 at 9:03
  • $\begingroup$ For $T1$, have a look at the basis for this topology, and note that a sequence converges to a point iff it is eventually in every basis set containing that point. $\endgroup$ Feb 13, 2020 at 9:13
  • $\begingroup$ For example the sequence 1/n does not converge to 0 in topology 4. $\endgroup$ Feb 13, 2020 at 10:02

1 Answer 1

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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 $a+\varepsilon$.

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