2. Preliminaries and a General Description of Free Boolean Topological Groups
All topological spaces and groups considered in this paper are assumed to be completely regular and Hausdorff.
The notation ω is used for the set of all nonnegative integers and for the set of all positive integers. By , we denote the group of order two. The cardinality of a set A is denoted by and the closure of a set A in an ambient topological space by . We denote the disjoint union of spaces X and Y by .
By a zero-dimensional space, we mean a space X with and by a strongly zero-dimensional space a space X with .
A Boolean group is a group in which all elements are of order two. Clearly, all Boolean groups are Abelian. Algebraically, all Boolean groups are free, because any Boolean group is a linear space over the field and must have a basis (a maximal linearly independent set) by Zorn’s lemma. This basis freely generates the given Boolean group. Moreover, any Boolean group (linear space) with basis X is isomorphic to the direct sum of copies of , i.e., the set of finitely supported maps with pointwise addition (in the field ). Of course, such an isomorphic representation depends on the choice of the basis.
A variety of topological groups is a class of topological groups closed with respect to taking topological subgroups, topological quotient groups and Cartesian products of groups with the product topology. Thus, the abstract groups
underlying the topological groups
G in a variety
V of topological groups (that is, all groups
without topology) form a usual variety
of groups. A variety
V of topological groups is full if any topological group
G for which
belongs to
V. The notions of a variety and a full variety of topological groups were introduced by Morris in [
8,
9], who also proved the existence of the free group of any full variety on any completely regular Hausdorff space
X.
Free objects of varieties of topological groups are characterized by the corresponding universality properties (we give a somewhat specific meaning to the word “universality,” but we use this word only in this meaning here). Thus, the free topological group on a space X admits the following description: X is topologically embedded in and, for any continuous map f of X to a topological group G, there exists a continuous homomorphism for which . As an abstract group, is the free group on the set X. The topology of can be defined as the strongest group topology inducing the initial topology on X. On the other hand, the free topological group is the abstract free group generated by the set X (which means that any map of the set X to any abstract group can be extended to a homomorphism of ) endowed with the weakest topology with respect to which all homomorphic extensions of continuous maps from X to topological groups are continuous. The free Abelian topological group on X, the free Boolean topological group on X and free (free Abelian, free Boolean) precompact groups are defined similarly; instead of continuous maps to any topological groups, continuous maps to topological Abelian groups, topological Boolean groups and precompact (Abelian precompact, Boolean precompact) groups should be considered.
There is yet another family of interesting varieties of topological groups. Following Malykhin (see also [
17]), we say that a topological group is linear if it has a base of neighborhoods of the identity element which consists of open subgroups. The classes of all linear groups, all Abelian linear groups and all Boolean linear groups are varieties of topological groups. These varieties are not full, but for any zero-dimensional space
X, there exist free groups of all of these three varieties on
X. Indeed, Morris proved that a free group of a variety of topological groups on a given space exists if this space can be embedded as a subspace in a group from this variety ([
8], Theorem 2.6). Thus, it suffices to embed any zero-dimensional
X in a Boolean linear topological group (which belongs to all of the three varieties under consideration). We do this below, but first we introduce more notation.
Whenever X algebraically generates a group G, we can set the length of the identity element to zero, define the length of any non-identity with respect to X as the least (positive) integer n such that for some and , , and denote the set of elements of length at most k by for ; then, . Thus, we use , and to denote the sets of words of length at most k in , and , respectively.
Now, we can describe the promised embedding.
Lemma 1. (i)
For any space X with , there exists a Hausdorff linear topological group such that is an algebraically free group on X, X is a closed subspace of , and all sets of words of length at most n are closed in .- (ii)
For any space X with , there exists a Hausdorff Abelian linear topological group such that is an algebraically free Abelian group on X, X is a closed subspace of , and all sets of words of length at most n are closed in .
- (iii)
For any space X with , there exists a Hausdorff Boolean linear topological group such that is an algebraically free Boolean group on X, X is a closed subspace of , and all sets of words of length at most n are closed in .
Proof. Assertion (i) was proven in [
22], Theorem 10.5. Let us prove (ii). Given a disjoint open cover
γ of
X, we set:
this is a subgroup of the free Abelian group on
X. We can assume that all words in
are reduced (if
is canceled with
, then
, because
and
γ is disjoint, and we can replace
by
). All such subgroups generate a group topology on the free Abelian group on
X; we denote the free Abelian group with this topology by
(we might as well take only finite covers).
The space X is indeed embedded in : given any clopen neighborhood U of any point , we have .
Let us show that
is closed in
for any
. Take any reduced word
with
, where
and
for
. Let
be clopen neighborhoods of
such that
and
are disjoint if
and coincide if
. We set:
Take any reduced word
in
and consider
. If, for some
, both
and
are canceled in
with some
and
, then, first,
(because any different letters in
g are separated by the cover
γ, while
and
must belong to the same element of this cover), and secondly,
(because
and
occur in
h with opposite signs). Hence,
, which contradicts
g being reduced. Thus, among any two letters
and
in
h, only one can be canceled in
, so that
cannot be shorter than
g. In other words,
.
The proof that X is closed in is similar: given any , we construct precisely the same γ as above (if ) or set (if ) and show that must contain at least one negative letter.
The Hausdorffness of is equivalent to the closedness of .
The proof of (iii) is similar. ☐
This lemma and Morris’ theorem cited above ([
8], Theorem 2.6) immediately imply the following theorem.
Theorem 2. For any space X with , the free, free Abelian and free Boolean linear topological groups , and are defined. They are Hausdorff and contain X as a closed subspace, and all sets , and are closed in the respective groups.
By definition, the free linear groups of a zero-dimensional space X have the strongest linear group topologies inducing the topology of X, that is, any continuous map from X to a linear topological group (Abelian linear topological group, Boolean linear topological group) extends to a continuous homomorphism from (, ) to this group.
Let X be a space, and let , , be its subspaces such that . Suppose that any is open in X if and only if each is open in (replacing “open” by “closed,” we obtain an equivalent condition). Then, X is said to have the inductive limit topology (with respect to the decomposition ). When talking about inductive limit topologies on , and , we always mean the decompositions , and and assume the sets , and to be endowed with the topology induced by the respective free topological groups.
For any space X, the free Abelian topological group is the quotient topological group of by the commutator subgroup, and the free Boolean topological group is the quotient of by the subgroup of squares (which is generated by all words of the form , ) (the universality of free objects in varieties of topological groups implies that the corresponding homomorphisms are continuous and open). Thus, is the image of (and of ) under a continuous open homomorphism.
The topology of free groups can be described explicitly. The first descriptions were given for free topological groups on compact spaces and free Abelian topological groups by Graev [
3,
4]; Tkachenko [
23,
24] and Pestov [
25] gave explicit descriptions of the topology of general free topological groups. There are also descriptions due to the author (see, e.g., [
26,
27]). Mal’tsev proposed a universal approach to describing the topology of free topological algebras, which is not quite constructive, but looks very promising [
7]. All descriptions of the topology of free and free Abelian topological groups of which the author is aware are given in [
22]. The descriptions of the free topological group topology are very complex (except in a few special cases); the topologies of free Abelian and Boolean topological groups look much simpler. Thanks to the fact that
, the descriptions of the free Abelian topological group topology given in [
22] immediately imply the following descriptions of the free topology of
.
I For each
, we fix an arbitrary entourage
of the diagonal of
in the universal uniformity of
X and set:
The sets
, where
ranges over all sequences of uniform entourages of the diagonal, form a neighborhood base at zero for the topology of the free Boolean topological group
.
II For each
, we fix an arbitrary normal (or merely open) cover
of the space
X and set:
The sets
, where Γ ranges over all sequences of normal (or arbitrary open) covers, form a neighborhood base at zero for the topology of
.
III For an arbitrary continuous pseudometric
d on
X, we set:
The sets
, where
d ranges over all continuous pseudometrics on
X, form a neighborhood base at zero for the topology of
.
It follows directly from the second description that the base of neighborhoods of zero in
(for zero-dimensional
X) is formed by the subgroups:
generated by the sets
with
γ ranging over all normal covers of
X. By definition, any normal cover of a strongly zero-dimensional space has a disjoint open refinement. Therefore, for
X with
, the covers
γ can be assumed to be disjoint, and for disjoint
γ, we have:
(see the proof of Lemma 1). A similar description is valid for the Abelian groups
(the pluses must be replaced by minuses). This leads to the following statement.
Proposition 3. For any strongly zero-dimensional space X and any , the topology induced on (on ) by (by ) coincides with that induced by (by ).
Proof. We can assume without loss of generality that n is even. Given any neighborhood U of zero in (in ), it suffices to take a sequence of disjoint covers such that and note that . ☐
Graev’s procedure for extending any continuous pseudometric
d on
X to a maximal invariant pseudometric
on
is easy to adapt to the Boolean case. Following Graev, we first consider free topological groups in the sense of Graev, in which the identity element is identified with a point of the generating space and the universality property is slightly different: only continuous maps of the generating space to topological groups
G that take the distinguished point to the identity elements of
G must extend to continuous homomorphisms [
3]. Graev showed that the free topological and Abelian topological groups
and
in the sense of Graev are unique (up to topological isomorphism) and do not depend on the choice of the distinguished point; moreover, the free topological group in the sense of Markov is nothing but the Graev free topological group on the same space to which an isolated point is added (and identified with the identity element).
The extension of a continuous pseudometric
d on
X to a maximal invariant continuous pseudometric
on the Graev free Boolean topological group
is defined by setting:
for any
. The infimum is taken over all representations of
g and
h as (reducible) words of equal lengths. The corresponding Graev seminorm
(defined by
for
, where 0 is the zero element of
) is given by:
The infimum is attained at a word representing g which may contain one zero (if the length of g is odd) and is otherwise reduced. Indeed, if the sum representing g contains terms of the form and , then these terms can be replaced by one term ; the sum does not increase under such a change thanks to the triangle inequality.
For the usual (Markov’s) free Boolean topological group
, which is the same as
(where 0 is an isolated point identified with zero), the Graev metric depends on the distances from the points of
X to the isolated point (they can be set to 1 for all
). The corresponding seminorm
on the subgroup
of
consisting of words of even length does not change. The subgroup
is open and closed in
, because this is the kernel of the continuous homomorphism
extending the constant continuous map
taking all
to 1. Thus, in fact, it does not matter how to extend
to
; for convenience, we set:
All open balls (as well as all open balls of any fixed radius not exceeding one) in all seminorms for d ranging over all continuous pseudometrics on X form a base of open neighborhoods of zero in .
Topological spaces X and Y are said to be M-equivalent (A-equivalent) if their free (free Abelian) topological groups are topologically isomorphic. We shall say that X and Y are B-equivalent if and are topologically isomorphic.
Given , we use to denote the topological subgroup of generated by Y.
A special role in the theory of topological groups and in set-theoretic topology is played by Boolean topological groups generated by almost discrete spaces, that is, spaces having only one non-isolated point. With each free filter on any set X, we associate the almost discrete space ( is a point not belonging to X); all points of X are isolated, and the neighborhoods of are , . For a space with infinitely many isolated points, there is no difference between the canonical definition of the groups , and and Graev’s generalizations , and . Indeed, Graev showed that and are unique (up to topological isomorphism) and do not depend on the choice of the distinguished point. Graev’s argument, which uses only the universality property, carries over word for word to free Boolean topological groups. Thus, when dealing with spaces associated with filters, we can identify with and assume that the only non-isolated point of is the zero of ; the descriptions of the neighborhoods of zero and the Graev seminorm are altered accordingly. To understand how they change, take the new (but in fact, the same) space , where 0 is one more isolated point, represent as the Graev free Boolean topological group with distinguished point (zero of ) 0, and consider the topological isomorphism between this group and the similar group with distinguished point (zero) .
For example, since any open cover of
can be assumed to consist of a neighborhood of
and singletons, Description II reads as follows in this case: For each
, we fix an arbitrary neighborhood
of
, that is,
, where
, and set:
The sets
, where the
W range over all sequences of neighborhoods of
, form a neighborhood base at zero for the topology of
. Strictly speaking, to obtain a full analogy with Description II of the Markov free group topology, we should set:
but this would not affect the topology: the former
equals the latter for a sequence of smaller neighborhoods, say
(remember that some of the
and
in the expression for
may equal
, that is, vanish).
Similarly, the base neighborhoods of zero in Description III take the form:
where
d ranges over continuous pseudometrics on
(again, we should set
, but this would not make any difference).
It is also easy to see that the isomorphism between (with distinguished point ) and does not essentially affect the sets of words of length at most n; in particular, they remain closed, and is the inductive limit of these sets with the induced topology if and only if has the inductive limit topology. In what follows, by , we shall usually mean the Graev free Boolean topological group with zero .
Thus, is naturally identified with the group of all finite subsets of X under the operation of symmetric difference (). The point , which is the zero element of , is identified with the empty set , which belongs to as the zero element. In the context of free Boolean groups on almost discrete spaces, we also identify each with the one-point set .
Sets of the form often arise in set-theoretic topology and in forcing. The role of X is often played by ω, and the filter is usually an ultrafilter with certain properties.
We assume all filters on ω to be free, i.e., to contain the Fréchet filter (of all cofinite sets).
A filter on ω is said to be a P-filter if, for any family of , , the filter contains a pseudo-intersection of this family, i.e., a set such that for all . For ultrafilters, this property is equivalent to being a P-point, or weakly selective, ultrafilter. A filter on ω is said to be a Ramsey filter if, for any family of , , the filter contains a diagonal of this family, i.e., a set such that, whenever and , we have . Ultrafilters with this property are known as Ramsey, or selective, ultrafilters.
We use the standard notation for the set of all finite subsets of ω and for the set of all finite sequences of elements of ω. Given , means that s is an initial segment of t, i.e., and all elements of are greater than all elements of s. For , by , we mean the greatest element of s in the ordering of ω. We also set .
5. Free Boolean Groups on Filters on ω
We have already seen in the preceding sections that free Boolean groups on almost discrete countable spaces (associated with filters on ω) exhibit interesting behavior. Moreover, they are encountered more often than it may seem at first glance.
Consider any Boolean group
with countable basis
X. As mentioned in
Section 2, this group is (algebraically) isomorphic to the direct sum (or, in topological terminology,
σ-product)
of countably many copies of
. There is a familiar natural topology on this
σ-product, namely the usual product topology; let us denote it by
. This topology induces the topology of a convergent sequence on
(where 0 denotes the zero element of
) and is metrizable; therefore, it never coincides with the topology
of the free Boolean topological group on
X. Moreover,
is contained in
only when
X is discrete or has the form
for some filter (recall that we assume all filters to be free,
i.e., contain the filter of cofinite sets, and identify the non-isolated points of the associated spaces with the zeros of their free Boolean groups). On the other hand, any countable space is zero-dimensional; therefore, any countable free Boolean topological group contains a sequence of subgroups with trivial intersection (see Theorem 2). In [
61], the following lemma was proven.
Lemma 16 ([
61])
. Let G be a countable non-discrete Boolean topological group that contains a family of open subgroups with trivial intersection. Then, there exists a basis of G such that the isomorphism taking this basis to the canonical basis of is continuous with respect to the product topology on . This immediately implies the following assertion.
Theorem 17. Any countable Boolean topological group containing a family of open subgroups with trivial intersection (in particular, any free Boolean topological or linear topological group on a countable space) has either a discrete closed basis or a closed basis homeomorphic to the space associated with a filter on ω.
Spaces of the form
are one of the rare examples where the free Boolean topological group is naturally embedded in the free and free Abelian topological groups as a closed subspace. The embedding of
into
is defined simply by
(for the Graev free groups, which are the same as Markov ones for such spaces), and the embedding into
is
, provided that
. These embeddings take
to:
and:
The topologies induced on
A and
F by
and
are easy to describe; the restrictions of base neighborhoods of the zero (identity) element to these sets are determined by sequences of open covers of
(
i.e., of neighborhoods of the non-isolated point
) in the same manner as in Description II (see [
22]). A straightforward verification shows that
A,
B and
are homeomorphic. The rigorous proof of this fact is rather tedious, and we omit it.
As mentioned in the Introduction, for any filter
, the free Boolean group on
is simply
. Any topology on
(as well as on any other set) is a partially ordered (by inclusion) family of subsets. Partial orderings of subsets of
have been extensively studied in forcing, and countable Boolean topological groups turn out to be closely related to them. In this section, we shall try to give an intuitive explanation of this relationship. The basic definitions and facts related to forcing can be found in Jech’s book [
64].
By a notion of forcing, we mean a partially ordered set (briefly, poset) . Elements of a notion of forcing are called conditions; given two conditions , we say that p is stronger than q if . A partially ordered set is separative if, whenever , there exists an which is incompatible with q. Thus, any topology is a generally non-separative notion of forcing, and the family of all regular open sets in a topology is a separative notion of forcing. Any separative forcing notion is isomorphic to a dense subset of a complete Boolean algebra. Indeed, consider the set for each . The family generates a topology on . The complete Boolean algebra mentioned above is the algebra of regular open sets in this topology.
Two notions of forcing and are said to be forcing equivalent if the algebras and are isomorphic or, equivalently, if can be densely embedded in and vice versa (which means that and give the same generic extensions).
Roughly speaking, given a countable transitive model of set theory, the method of forcing extends by adding a so-called generic subset (called also a generic filter) G of not belonging to ; the extended model, called a generic extension of , contains , which has certain desired properties ensured by the choice of and G.
In the context of free Boolean groups on filters, most interesting are two well-known notions of forcing, Mathias forcing and Laver forcing relativized to (usual) filters on ω.
In Mathias forcing relative to a filter , the forcing poset, denoted , is formed by pairs consisting of a finite set and an (infinite) set such that every element of s is less than every element of A in the ordering of ω. A condition is stronger than () if , and .
The poset in Laver forcing consists of subsets of the set of ordered finite sequences in ω. However, it is more convenient for our purposes to consider its modification consisting of subsets of . Thus, we restrict the Laver forcing poset to the set of strictly increasing finite sequences in ω (this restricted poset is forcing equivalent to the original one) and note that the latter is naturally identified with . Below, we give the definition of the corresponding modification of Laver forcing.
The definition of Laver forcing uses the notion of a Laver tree. A Laver tree is a set
p of finite subsets of
ω such that:
- (i)
p is a tree (i.e., if , then p contains any initial segment of t),
- (ii)
p has a stem, i.e., a maximal node , such that or for all and
- (iii)
if and , then the set is infinite.
In Laver forcing relative to , the poset, denoted , is the set of Laver trees p such that for any with , ordered by inclusion.
The Mathias and Laver forcings
and
have the special feature that they diagonalize the filter
(
i.e., add its pseudo-intersection). They determine two natural topologies on
: the Mathias topology
generated by the base:
and the Laver topology
generated by all sets
such that:
It is easy to see that the Mathias topology is nothing but the topology of the free Boolean linear topological group on (recall that linear groups are those with topology generated by subgroups): a base of neighborhoods of zero is formed by the sets with , that is, by all subgroups generated by elements of .
The neighborhoods of zero in the Laver topology are not so easy to describe explicitly; their recursive definition immediately follows from that given above for general open sets (the only condition that must be added is ). Thus, U is an open neighborhood of zero if, first, ; by definition, U must also contain all for some (moreover, U may contain no other elements of size one); for each of these n, there must exist an such that and U contains all with (moreover, U may contain no other element of size two); for any such (), there must exist an such that and U contains all with , and so on. Thus, each neighborhood of zero is determined by a family of elements of . Clearly, the topology is invariant with respect to translation by elements of ; upon a little reflection, it becomes clear that is the maximal invariant topology on in which the filter converges to zero. (An invariant topology is a topology with respect to which the group operation is separately continuous; groups with an invariant topology are said to be semi-topological. The convergence of to zero means that induces the initially given topology on .) Since the free group topology is invariant as well, it is weaker than .
The Mathias topology is, so to speak, the uniform version of the Laver topology: a neighborhood of zero in the Laver topology determined by a family
is open in the Mathias topology if and only if there exists a single
such that
for each
s. (In [
65], the corresponding relationship between Mathias and Laver forcings was discussed from a purely set-theoretic point of view.) Hence,
.
The topology of the free Boolean topological group on occupies an intermediate position between the Mathias and the Laver topology: it is not so uniform as the former, but more uniform than the latter. A neighborhood of zero is determined not by a single element of the filter (like in the Mathias topology), but by a family of elements of assigned to (like in the Laver topology), but these elements depend only on the lengths of s.
The following theorem shows that the Laver topology is a group topology only for special filters. This theorem was proven in 2007 by Egbert Thümmel, who kindly communicated it, together with a complete proof, to the author. The symbols and in its statement denote the topology of the free topological group and the inductive limit topology of , respectively.
Theorem 18 (Thümmel, 2007 [
66])
. For any filter on ω, the following conditions are equivalent:- (i)
is Ramsey;
- (ii)
;
- (iii)
is a group topology;
- (iv)
for any sequence of , , the set is open in .
This theorem is particularly interesting because its original (Thümmel’s) proof uses an argument that is simple and still quite typical of the method of forcing. The proof given below only slightly differs from Thümmel’s and uses this argument, as well.
Proof. First, note that . Indeed, the first two inclusions are obvious, and the third one follows from Proposition 3 (or from the inclusion noted above) and the observation that is the inductive limit of its restrictions to .
Thus, to prove the implication (i) ⇒ (ii), it suffices to show that
for any Ramsey filter. Let
U be a neighborhood of
in
. For each
, we set:
Since the number of with is finite, it follows that . Take a diagonal for the family . We can assume that . Clearly, , whence .
The implication (ii) ⇒ (iii) is trivial.
Let us prove (iii) ⇒ (iv). Note that it follows from (iii) that , because and is the strongest group topology inducing the initially given topology on . It remains to note that any set of the form , where , is open in .
We proceed to the last implication (iv) ⇒ (i). Take any family and consider the set U defined as in (iv). Since this is an open neighborhood of zero in the group topology , there exists an open neighborhood V of zero (in ) such that . The set belongs to (because induces the initially given topology on and is a diagonal of . ☐
This theorem is worth comparing to Judah and Shelah’s proof that if
is a Ramsey ultrafilter, then
is forcing equivalent to
([
67], Theorem 1.20 (i)).
Thümmel also obtained the following remarkable result as a simple corollary of Theorem 18.
Theorem 19 (Thümmel, 2007 [
66])
. Given a filter on ω, the group is extremally disconnected if and only if is a Ramsey ultrafilter. Proof. The proof of the if part is essentially contained in Sirota’s construction of a (consistent) example of an extremally disconnected group [
56]. The proof of the only if part is based on the equivalence (iv) ⇔ (i) of Theorem 18: for any family
, the set
is open even in the Mathias topology, and its closure in
, which must be open by virtue of extremal disconnectedness, is
. The assertion (iv) ⇔ (i) implies that
is a Ramsey filter. It remains to apply Theorem 13 and recall that
is extremally disconnected if and only if
is an ultrafilter. ☐
Thümmel has never published these results, and Theorem 19 was rediscovered by Zelenyuk, who included it, among other impressive results, in his book [
59] (see Theorem 5.1 in [
59]).
Combining Theorem 19 with Corollary 15, we obtain yet another corollary.
Corollary 20. The free Boolean group on a non-discrete countable space X is extremally disconnected if and only if X is an almost discrete space associated with a Ramsey ultrafilter.
Free Boolean topological and free Boolean linear (that is, Mathias) topological groups on spaces associated with filters, as well as Boolean groups with other topologies determined by filters, are the main tool in the study of topological groups with extreme topological properties (see [
59] and the references therein). However, free Boolean (linear) topological groups on filters arise also in more “conservative” domains. We conclude with mentioning an instance of this kind.
The most elegant (in the author’s opinion) example of a countable non-metrizable Fréchet–Urysohn group was constructed by Nyikos in [
68] under the relatively mild assumption
(Hrušák and Ramos-García have recently proven that such an example cannot be constructed in ZFC [
69]).
It is clear from general considerations that test spaces most convenient for studying convergence properties that can be defined pointwise (such as the Fréchet–Urysohn property and the related -properties) are countable almost discrete spaces (that is, spaces of the form ), and the most convenient test groups for studying such properties in topological groups are those generated by such spaces, simplest among which are free Boolean linear topological groups. Thus, it is quite natural that Nyikos’ example is for a very cleverly constructed filter . In fact, he constructed it on (which does not make any difference, of course) as the set of neighborhoods of the only non-isolated point in a Ψ-like space defined by using graphs of functions from a special family. In the same paper, Nyikos proved many interesting convergence properties of groups for arbitrary filters on ω. We do not give any more details here: the interested reader will gain much more benefit and pleasure from reading Nyikos’ original paper.