Clarification on the limit superior and inferior in topological spaces

Question

Let $(X, \mathcal{T})$ be a topological space and $\left(A_n\right)_{n\in \mathbb{N}}$ a sequence of subsets of $X$. My reading material defines the limit superior as the set of points of $X$ such that every neighborhood of a point $x \in \lim\sup_nA_n$ intersects with infinitely many members of the sequence $\left(A_n\right)_{n\in \mathbb{N}}$. The limit inferior is defined similarly as a set of points of $X$, but such that any neighborhood of a $x \in \lim\inf_nA_n$ intersects with all except for finitely many members of the sequence $\left(A_n\right)_{n\in \mathbb{N}}$.

Then with these definitions, are the complements of $ \lim\sup_nA_n$ and $\lim\inf_nA_n$ sets $S_1$ and $S_2$, respectively, such that 1.) any $x\in S_1$ has a neighborhood that intersects with at most finitely many members of the sequence $\left(A_n\right)_{n\in \mathbb{N}}$ and 2.) formulated logically $\forall x \in S_2:\forall x \in U \in \mathcal{T}:\forall n \in \mathbb{N}:\exists n_0 \geq n: U\cap A_n = \varnothing$?

@Kavi Rama Murthy: Limit infeiror and limit superior are purely set theorertic concepts --- The OP is asking about what is often called (now, at least; older literature varies a lot) the Kuratowski lim-inf and lim-sup, which are extensively treated in Kuratowski's 2-volume topology treatise. See Kuratowski lim sup and lim inf of a sequence of subsets of a topological space? AND lim sup and lim inf of a sequence of subsets relative to a topology and questions linked to this second question. — Dave L. Renfro, Feb 25 '22 at 12:23

score 1 · Answer 1 · answered Feb 25 '22 at 14:07

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So $x \in \limsup_n A_n$ iff

$$\forall U \in \mathcal{N}(x): \exists_\infty n: U \cap A_n \neq \emptyset\tag{1}$$

which has as negation

$$\exists U \in \mathcal{N}(x): \forall_\infty n: U \cap A_n = \emptyset\tag{2}$$

where $\forall_\infty$ means "for all but finitely many". The same logic rules apply to $\exists_\infty$ (there exist infinitely many) and this $\forall_\infty$ quantifier regarding negation as usual $\forall, \exists$ with regards to negations, as you see illustrated above (when these $n$ range over $\Bbb N$). (This is standard fare in mathematical logic)

So $(2)$ describes $\left(\limsup_n A_n\right)^\complement$.

The $\liminf_n A_n$ points obey exactly

$$\forall U \in \mathcal{N}(x): \forall_\infty n: U \cap A_n \neq \emptyset\tag{3}$$ with complementary points obeying:

$$\exists U \in \mathcal{N}(x): \exists_\infty n: U \cap A_n = \emptyset\tag{4}$$

answered Feb 25 '22 at 14:07

Henno Brandsma

242,131

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For those interested in a little more about these quantifiers, see my answer to Why haven't mathematicians come up with an efficient way of writing “sufficiently”, e.g. “for $n$ sufficiently large”, and for applications (outside of mathematical logic) to a natural generalization of their use of "finite", see my answer to Relations between the "$\forall$" quantifier and integration. – Dave L. Renfro Feb 25 '22 at 17:48
@DaveL.Renfro nice! I know the category and measure ones from Kechris’ nice book. It’s a useful addition for reasoning about some facts. And compact formulations as well. – Henno Brandsma Feb 25 '22 at 18:04
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There's also the following two sci.math posts, which might get missed by those in a hurry. The first is cited in my first of the above-cited MSE answers (and contains some references at the end not in either of my MSE answers), and the second is cited (without a link) in the first sci.math post: 21 December 2004 post in the sci.math thread Generalized Quantifiers AND 20 December 2004 post in the sci.math thread What is meant by a meager subset??. – Dave L. Renfro Feb 25 '22 at 18:58

Clarification on the limit superior and inferior in topological spaces

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