A Survey of Cardinal Invariants of Topological Groups

(This is a presentation I gave for Bill Weiss’ course Set-Theoretic Topology on April 19, 2013. In class we discussed some cardinal invariants and how they are related; here I will survey what happens when we look at the cardinal invariants of topological groups.)

This review follows very closely the discussion in section 5.2 of Arhangel’skii and Tkachenko’s book “Topological Groups and Related Structures“. Another good resource is section 3 of Comfort’s article “Topological Groups” in the Handbook of Set-Theoretic Topology. The only thing I claim to be my own are the (unsourced) pictures I have provided.

Conventions and Notation

By convention, \mathcal{N} will be the set of all open neighbourhoods of the identity in some fixed topological group G . The identity will be written as e , and we use multiplicative notation for our groups. Whenever relevant, we assume that G is at least T_1 (hence also regular).

Recall that the following standard facts about topological groups G with neighbourhood basis \mathcal{N} at the identity:

  1. (\forall U \in \mathcal{N})(\exists V \in \mathcal{N})[VV^{-1} \subseteq U] ;
  2. (\forall a \in U \in \mathcal{N})(\exists V \in \mathcal{N})[aV \subseteq U] ;
  3. (\forall U \in \mathcal{N})(\forall x \in G)(\exists V \in \mathcal{N})[xVx^{-1} \subseteq U] .
prestuff 1_1
The basic good properties of topological groups

Recall the following cardinal invariants for a topological space (X, \tau) which are all assumed to be infinite:

  • w(X) := \min \{\, \vert \mathcal{B} \vert : \mathcal{B} is a basis for X \,\} , is the weight of X ;
  • d(X) := \min \{\, \vert D\vert : \overline{D} = X\,\} , is the density of X ;
  • \chi(p, X) := \min \{\, \vert \mathcal{B} \vert : \mathcal{B} is a local base at p \in X \,\} , is the local character at p ;
  • \chi(X) := \sup_{p \in X} \chi(p, X) , is the character of X ;
  • \pi w(X) and \pi \chi(X) , the \pi -weight and \pi -character (respectively), are defined analogously to weight and character, expect we do not ask “that the base point be in the open set”;
  • L(X) := \min \{\, \kappa : every cover of X has a subcover of cardinality \leq \kappa \,\} , is the Lindelof number of X ;
  • e(X) := \sup\{\, \vert D \vert : D  is closed and discrete \,\} , is the extent of X ;
  • c(X) := \sup\{\, \vert \mathcal{U} \vert: \mathcal{U} is a pairwise disjoint family of open sets in X \,\} , is the cellularity of X .

You can see the relations here, from the picture taken from Hodel’s Article in the Handbook of Set-Theoretic Topology.

Fig. 1. (Hodel's article from the Handbook of Set-Theoretic Topology)
Fig. 1. (Hodel’s article from the Handbook of Set-Theoretic Topology)

A cardinal invariant for topological groups

Definition 1. A topological group is said to be \kappa -narrow iff \forall U \in \mathcal{N} there is an A \in [G]^{\kappa} such that A \cdot U = G . The index of narrowness, ib(G) , is the minimal infinite cardinal \kappa such that G is \kappa -narrow.

Facts. For a topological group G :

  1. ib(G) \leq e(G) and hence ib(G) \leq L(G) \leq w(G) ;
  2. ib(G) \leq c(G) ;
  3. w(G) = ib(G) \cdot \chi(G) .

proof of 1. Fix \kappa := e(G) . All we need to show is that given a (wlog) open symmetric U \in \mathcal{N} , we can translate it at most \kappa -times and cover all of G .

Suppose for the sake of contradiction that this does not hold for a particular U . By recursion, define \{x_\alpha : \alpha < \kappa^+\} so that x_\alpha \notin x_\beta U for all \alpha < \beta < \kappa^+ .

Now take a V \in \mathcal{N} such that V^4 \subseteq U . The family \{x_\alpha V : \alpha < \kappa\} witnesses the fact that \{x_\alpha : \alpha < \kappa^+\} is closed and discrete. This is a closed discrete family of cardinality \kappa^+ , a contradiction. [QED]

fact 1_1
Used in the proof of 1.

proof of 2. Let U \in \mathcal{V} . Choose a symmetric V \in \mathcal{N} such that V^2 \subseteq U . Take a family A that is maximal with respect to the property: \forall a,b \in A we have aV \cap bV = \emptyset .

Consider \{aV : a \in A\} , which is a family of disjoint open sets of cardinality \vert A \vert \leq c(G) .

Claim: AU = G .

Let x \in G . By maximality of A , there is an a \in A such that xV \cap aV \neq \emptyset . So then x \in aVV^{-1} \subseteq aU . [QED]

fact 1_2
Used in the proof of 2.

proof of 3. By (1) we know that ib(G) \leq w(G) , and it is clear that \chi(G) \leq w(G) . We only need to show that w(G) \leq \chi(G) \cdot ib(G) .

Set \kappa := \chi(G) \cdot ib(G) , and let \mathcal{H} be a basis at the identity such that \vert \mathcal{H} \vert \leq \kappa . By narrowness, for all U \in \mathcal{H} there is an S_U \in [G]^{\kappa} so that S_U \cdot U = G . Set \mathcal{B} := \{\, xU : U\in \mathcal{H}, x \in S_U \,\} , which clearly has cardinality \leq \kappa .

Claim: \mathcal{B} is a basis for G .

Let a \in O be an open neighbourhood in G . There exist U,V \in \mathcal{H} such that aU \subseteq O and V^{-1}\cdot V \subseteq U . We know that S_V \cdot V = G , so there is an x \in S_V such that a \in xV (and so also x \in a V^{-1} ).

Thus xV \subseteq a V^{-1} V \subseteq a U \subseteq O . [QED]

fact 1_3
Used in the proof of 3.

What are some special cases of this?

Corollary. If G is an infinite precompact group, then w(G) = \chi(G) .

Usual Cardinal invariants in topological groups

Recall that for any topological group ib(G) \leq c(G) \leq d(G) and ib(G) \leq L(G) \leq k(G) . So from the previous facts, we get the following relations which do not hold for the class of compact Hausdorff spaces. The two-arrows space – i.e. [0,1]\times\{0,1\} with the lexicographical ordering- is an example of a compact, first-countable, separable Hausdorff space with uncountable weight.

Fact: For any topological group G :

  1. w(G) \leq d(G) \cdot \chi(G) ;
  2. w(G) \leq k(G) \cdot \chi(G) ;
  3. w(G) \leq L(G) \cdot \chi(G) .

Proposition: For a topological group G

  1. \chi (G) = \pi \chi(G) ;
  2. w(G) = \pi w(G) ;

proof of 1. We need only show \chi (G) \leq \pi \chi(G) . In fact, since we are checking the character for a topological group, we need only check the local character at the identity.

Fix \kappa := \pi \chi(G) , and let \mathcal{B} be a \pi -base at the identity of cardinality \kappa .

Claim: \mathcal{C} := \{\, BB^{-1} : B \in \mathcal{B}\,\} is a local base at the identity (which clearly has cardinality \leq \kappa ).

Let U \in \mathcal{N} , and find a V \in \mathcal{N} such that VV^{-1} \subseteq U . Since \mathcal{B} is a \pi -base, there is a B \in \mathcal{B} such that B \subseteq V .

Thus e \in BB^{-1} \subseteq VV^{-1} \subseteq U , and BB^{-1} \in \mathcal{C} . [QED]

fact 2_1
Used in the proof of 1.

proof of 2. We recall the obvious inequality \pi w(G) \leq w(G) , and we have already done all of the work for the other direction:

w(G) \leq d(G) \cdot \chi(G) = d(G) \cdot \pi \chi(G) = \pi w(G) .

The first inequality is by Fact X.1, the second equality is by part a, and the last equality is obvious. [QED]

Compact Groups

We can combine these results with the classical results (from class) to get the following relations:

Proposition. Let G be an infinite compact topological group. Then:

  1. \pi \chi(G) = \chi(G) = w(G) = \pi w(G) ;
  2. \vert G \vert = 2^{w(G)} ;
  3. d(G) = ln(w(G)) , where ln(w(G)) is the least cardinal \kappa such that w(G) \leq 2^{\kappa} .

Recall:

Cech-Pospisil Theorem. (7.19 in Hodel’s article in the Handbook of Set-Theoretic Topology). Let X be a compact space such that for each p \in X the local character \chi (p, X) \geq \kappa . Then 2^\kappa \leq \vert X \vert .

proof of 1. This follows from what we have already done, namely that w(G) = ib(G) \cdot \chi(G) , and ib(G) \leq l(G) .

proof of 2. For a topological group, each point has local character equal to the group’s character. So the C-P theorem gives us that 2^{\chi(G)} \leq \vert G \vert . For the other direction, we note that every Hausdorff space satisfies \vert X \vert \leq 2^{w(X)} . [QED]

 

Some hard simple facts

Here are three big theorems related to topological groups:

Theorem [Birkhoff-Kakutani, 1936]. A topological group is metrizable if and only if it is first-countable.

Theorem [Haar, 1933]. Every locally compact topological group admits a (unique up to a constant) translation invariant (Haar) measure.

This theorem can often be used to simplify proofs involving locally compact groups. The existence of a measure that is positive on open sets usually gives a lot of understanding of the group.

Theorem [Tkachenko, 1983]. Every \sigma -compact topological group satisfies the countable chain condition.

Contrast this with \aleph_\omega with the linear-order topology which is a \sigma -compact space with a very large uncountable collection of disjoint open sets. Also, \aleph_\omega \cup \{\, \aleph_\omega\,\} is a compact space that isn’t ccc.

 

References

Arkhangelʹskiĭ, A. V., & Tkachenko, M. (2008). Topological groups and related structures. Amsterdam: Atlantis Press.

Kunen, K., & Vaughan, J. E. (1984). Handbook of set-theoretic topology. Amsterdam: North Holland.

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