# For the given topologies, which sequences converge to which limits?

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.

• 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. Feb 13, 2020 at 9:03
• 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. Feb 13, 2020 at 9:13
• For example the sequence 1/n does not converge to 0 in topology 4. Feb 13, 2020 at 10:02

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 $$a+\varepsilon$$.