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This post is primarily a reference request.

In combinatorics and other areas, we use filter quantifiers to simplify the statements of various definitions, theorems and proofs. The general idea is that, if $mathcal{F}$ is a filter on a set $W$, and $P(x)$ is a property that an element of $W$ may or may not possess, then $(forall_mathcal F,x) P(x)$ means that $P$ holds on a set of elements that is in $mathcal F$. This quantifier is also sometimes written $(mathcal F, x) P(x)$. There is also a dual quantifier, $(exists_mathcal F , x) P(x)$, which says that the set of elements satisfying $P$ is stationary in $mathcal F$.

Specific examples include:

• If we let $mathcal F$ be the Fréchet filter of cofinite subsets of $mathbb N$, we obtain the infinitary quantifiers $forall^infty$ and $exists^infty$ ("large" sets are cofinite, "small" sets are finite, and stationary sets are infinite).




• If we let $mathcal F$ be the filter of measure 1 subsets of $[0,1]$, we obtain a kind of measure quantifier ("large" sets have measure 1, "small" sets have measure 0, and stationary sets have positive measure).

I am interested in finding any undergraduate textbooks, or any general logic textbooks, that discuss these filter quantifiers in detail. The only references I have been able to locate are graduate-level papers on combinatorics. There are a few isolated internet posts, such as 1 and 2. It would be nice to have something to point a younger student towards.

I am also interested in the following question. Let $mathcal F$ be the Fréchet filter of cofinite subsets of $mathbb N$, let $(a_n)$ be a sequence of real numbers and let $z$ be a real number. For each open interval $I$, let $P_I = { n in mathbb N : a_n in I}$. Then the usual definition of convergence can be restated as: $(a_n)$ converges to $z$ if and only if $(forall_mathcal F, x) P_I(x)$ holds for every open interval $I$ containing $z$ (this could also be written $(forall^infty,x)P_I(x)$). And $z$ is a cluster point of $(a_n)$ if and only if $(exists_mathcal F, x) P_I(x)$ for every open interval $I$ containing $z$.

We can generalize the usual notion of convergence by simply replacing the Fréchet filter $mathbb F$ with any other filter on $mathbb N$. I am interested in any references about this generalization.

I would suspect there should be a real analysis text that discusses this alternate notion of convergence, at least in exercises. Please note that this is not prima facie the same as the notion of filter convergence in general topology, although comments by Alex Kruckman below show there is a relationship. Since I asked this originally, I've learned from this Tricki post that this method can be used to construct Banach limits, using the method described below (instead of the Hahn-Banach theorem, which is how I had seen it). I would be interested in any other interesting examples of what can be done with this sort of generalized "convergence on a filter". Perhaps there are other Hahn-Banach type results that can be converted to use ultrafilters.

We can generalize the usual notion of convergence by simply replacing the Fréchet filter $mathbb F$ with any other filter on $mathbb N$. I am interested in any references about this generalization.

My answer concerns one specific application of this convergence in general topology and topological algebra. Namely, given an (ultra)filter $mathcal F$ on $Bbb N$, a topological space $X$ (remark that we don't assume any separation axioms in our consideration of the definitions), and a sequence ${x_n}_{ninBbb N}$ of points of $X$, we say that a point $xin X$ is a $mathcal F$-limit of ${x_n}$ provided for any neighborhood $U$ of $x$ a set ${ninBbb N: x_nin U}$ belongs to $mathcal F$. This notion is known for a long time and was considered, for instance, by Bernstein [Ber], Furstenberg [Fur, p. 179], and Akin [Aki, p. 5, 61].

By $Bbb N^*$ we denote the set of all free ultrafilters on $Bbb N$. In this topic I am mainly acquainted with $mathcal F$-compact spaces for given $mathcal FinBbb N^*$, that is such spaces $X$ in which any sequence has a $mathcal F$ limit [Vau, Def. 4.5]. These spaces are important by the following two reasons.

Stratification of compact-like spaces. In general topology are well-studied different classes of compact-like spaces and relations between them, see, for instance, basic [Eng, Chap. 3] and general works [DRRT], [Mat], [Vau], [Ste], [Lip]. The including relations between classes are often visually represented by arrow diagrams, see, [Mat, Diag. 3 at p.17], [DS, Diag. 1 at p. 58] (for Tychonoff spaces), [Ste, Diag. 3.6 at p. 611], and [GR, Diag. at p. 3].

Recall that a space is countably compact iff each its open countable cover of has a finite subcover iff each its infinite subset $A$ has an accumulation point $x$ (the latter means that each neighborhood of $x$ contains infinitely many points of the set $A$). Clearly, if a space $X$ is $mathcal F$-compact for some $mathcal FinBbb N^*$ then $X$ is countably compact (see [Vau, Lemma 4.6] for $X$ which are $T_3$).

We may call a space $X$ ultracompact if X is $mathcal F$-compact for each $mathcal Fin Bbb N^*$. But it can be shown that each $omega$-bounded space is ultracompact, and, conversely, by [Theorem 4.9, Vau] each ultracompact $T_3$ space is $omega$-bounded. Recall that a space is $omega$-bounded, if each its countable subset has compact closure.

Thus for any utrafilter $mathcal Finmathbb N^*$ a class of $mathcal F$-compact spaces is intermediate between the classes of $omega$-bounded and countably compact spaces. An other intermediate class between these two classes is a class of totally countably compact spaces, that is spaces in which any sequence contains a subsequence with compact closure. There exists a Frolík class $mathcal C$ of spaces, which is intermediate between totally countably compact and $mathcal F$-compact for Tychonoff spaces, for an arbitrary, but fixed ultrafilter $mathcal FinBbb N^*$ (see the class Definition 3.9 in [Vau] (in which, I guess is missed that $Y$ is countably compact), the class characterization in [Vau, Theorem 3.11], and the diagram at [Vau, p.572]).

Given an utrafilter $mathcal Finmathbb N^*$, it is easy to check that a product of any family of $mathcal F$-compact spaces is $mathcal F$-compact. On the other hand, it is easy to show that if the $2^frak c$-th power of a space is countably compact, then the space is $mathcal F$-compact for some $mathcal FinBbb N^*$.

A background of these results is the investigation of productivity of compact-like spaces, motivated by the fundamental Tychonoff theorem, stating that a product of a family of compact spaces is compact. On the other hand, there are two countably compact spaces whose product is not feebly compact (see [Eng, the paragraph before Theorem 3.10.16]). Recall that a space $X$ is pseudocompact, if each continuous real-valued function on $X$ is bounded. Clearly, each countably compact space is pseudocompact. A subclass of countably compact spaces is constituted by sequentially compact spaces. Recall that a space is sequentially compact, if each sequence of its points has a convergent subsequence. The product of a countable family of sequentially compact spaces is sequentially compact [Eng, Theorem 3.10.35]. But already the Cantor cube ${0,1}^frak c$ is not sequentially compact (see [Eng, the paragraph after Example 3.10.38]). On the other classes of compact-like spaces preserved by products, see [Vau, $S$ 3-4] (especially Theorem 3.3, Proposition 3,4, Example 3.15, Theorem 4.7, and Example 4.15) and $S$7 for the history, and [Ste, $S$ 5].

The Wallace problem. A semigroup $(S,cdot)$ is cancellative if for any $a,bin S$ we have $a=b$, provided there exists $xin S$ such that $ax=bx$ or $xa=ab$. A semigroup $(S,cdot)$ endowed with a topology is called a topological semigroup, if the multiplication map $Stimes Sto S$ is continuous. If, moreover, $S$ is a group then $S$ is called a paratopological group. A paratopological group $G$ such that the inverse map $xmapsto x^{-1}$ is continuous is a topological group.

The proof of [Tom$_3$, Theorem 3.1 in a preprint] shows that for each $mathcal FinBbb N^*$ each $mathcal F$-compact cancellative Hausdorff topological semigroup $S$ is group. That is, $S$ is a paratopological group. Since $Stimes S$ is countably compact, it is a topological group, see [RR]. Tomita also noted that we can modify proof of Theorem 3.1 to show that each sequentially compact cancellative semigroup $S$ is a group. Bokalo and Guran [BG, Theorem 6] also showed that each sequentially compact Hausdorff cancellative topological semigroup is a topological group.

A background of these results is the following well-known in topological algebra and still open under axiomatic assumptions problem [Wal], [Com$_2$] posed by Wallace in 1953. He remarked that several authors proved that a Hausdorff compact cancellative topological semigroup is a topological group and asked: whether every countably compact cancellative topological semigroup $S$ a topological group?

In 1957 Ellis [Ell] showed that a locally compact regular group even with separately continuous operation is a topological group (for a paratopological group we can drop regularity, see Proposition in this answer). In 1972, Mukherjea and Tserpes [MT] showed that the answer is affirmative for first countable semigroups. In 1993 Grant [Gra$_2$] obtained affirmative answers for particular cases. He also mentioned that it was known that the answer is affirmative for $omega$-bounded semigroups.

On the other hand, in 1996 Robbie and Svetlichny [RS] showed under CH that there is a counterexample for the Wallace Problem. Tomita [Tom$_3$] showed that there is a counterexample to the Wallace Problem under $ MA_{countable}$. (Recall that $MA_{countable}$ is Martin's Axiom restricted to countable posets. This axiom is equivalent to a strong form of the Baire Category Theorem: a circle $Bbb T$ is not a union of less than continuum many closed nowhere dense sets).

Now we have a following framework for counterexamples to the Wallace problem. Namely, let TT be the following axiomatic assumption: there is an infinite torsion-free abelian countably compact topological group without non-trivial convergent sequences. The first example of such a group constructed by Tkachenko under the Continuum Hypothesis [Tka]. Later, the Continuum Hypothesis weakened to the Martin Axiom for $sigma$-centered posets by Tomita in [Tom$_3$], for countable posets in [KTW], and finally to the existence continuum many incomparable selective ultrafilters in [MuT]. Yet, the problem of the existence of a countably compact group without convergent sequences in ZFC seems to be open, see [DiS].

The proof of [BDG, Lemma 6.4] implies that under TT there exists a Hausdorff group topology on a free abelian group $F$ generated by the set $frak c$ such that for each countable infinite subset $M$ of the group $F$ there exists an element $alpha in overline Mcapfrak c$ such that $Msubsetlangle alpha rangle$. Then the free abelian semigroup generated by the set $frak c$ is countably compact, but not a group. Also we can modify a topology on $F$ making it a Hausdorff countably compact paratopological non-topological group, see [Rav, Example 2]. Conversely, by [Rav, Proposition 8], each such a group contains an element $g$ such that the closure $overline{langle grangle}$ of a cyclic subgroup $langle grangle$ generated by an element $g$ is not a group. I recall that Pfister [Pfi] showed that each regular (locally) countably compact paratopological group is a topological group. Banakh et al. in [BBGGR] proved a bit more stronger result: a subgroup of a locally countably compact $T_3$ topological semigroup with open left shifts is a topological group.

As far as I know, a question whether there exists under ZFC a cancellative Hausdorff topological semigroup $S$ such that $S$ (or $Stimes S$) is countably compact, but $S$ is not a group, is still unanswered.

See the surveys [Com$_1$], [Com$_2$] and [Chr] for a discussion on the Wallace problem [GKO], [Gra$_1$], [Gra$_2$], [Hel], [MT], [RS], [Tom$_1$]-[Tom$_6$], [Wal], [Yur] for other results on it.

Situation with counterparts of the Wallace problem with another compact-like properties is the following.

For pseudocompact case there is a following relatively simple counterexample. Let $H=prod_{alphain A} H_alpha$ be a Tychonoff product of an uncountable family of compact topological groups, such that for each $alphain A$ a group $H_alpha$ contains a non-periodic element $h_alpha$. For instance, for each $alpha$ we can take as $H_alpha$ the circle group $Bbb T$. Let $G={(g_alpha)in H$ such that $g_alpha=e$ for all but countably many $alphain A}$, $S$ be a subsemigroup of $H$ generated by $G$, and $h=(h_alpha)$. Since $h^nnotin G$ for any natural $n$, $h^{-1}notin S$, thus $S$ is not a group. Since $S$ contains a dense countably compact subset $G$, it is pseudocompact. Also there are feebly compact Hausdorff paratopological non-topological groups, see, for instance [Rav, Example 3] or [ST, Theorems 1 and 2]. On the other hand, there are many conditions providing that a feebly compact paratopological group is a topological group, see, for instance, [Rav, Proposition 3].

Given a cardinal $kappa$, a space is $kappa$-compact if every its open cover of size $kappa$ has a finite subcover. We recall that Kunen's Axiom (KA) assumes that there exists a ultrafilter $mathcal FinBbb N^*$ with a basis of size $omega_1$. KA trivially follows from CH, but also KA $&$ $frak c=aleph_2$ is consistent, see [Kun, Ex. VIIIA.10]. In [Tom$_3$] Tomita proved that under KA each Tychonoff $omega_1$-compact cancellative topological semigroup is a group. On the other hand, he showed that under MA$_{countable}$ $&$ $operatorname{cf}frak c>omega_1$ a topological group $Bbb T^{frak c}$ contains an $omega_1$-compact subsemigroup, which is not a group. Remark that MA$_{countable}$ also is consistent with $frak c = aleph_2$ [Tom$_3$]. Therefore the answer to the question for Tychonoff $omega_1$-compact semigroups is independent on $frak c = aleph_2$.

References

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