On fundamental groups of quotient spaces

In classical covering space theory, a covering map induces an injection of fundamental groups. This paper reveals a dual property for certain quotient maps having connected fibers, with applications to orbit spaces of vector fields and leaf spaces in general.     

On subgroups of finite index in compact Hausdorff groups

Let G be a compact Hausdorff group and n a positive integer. It is proved that all subnormal subgroups of G of index dividing n are open if and only if there are only finitely many such subgroups, and that all subgroups of finite index in G are open if and only if there are only countably many such subgroups.     

On the fundamental groups of one-dimensional spaces

We study here a number of questions raised by examining the fundamental groups of complicated one-dimensional spaces. The first half of the paper considers one-dimensional spaces as such. The second half proves related results for general spaces that are needed in the first half but have independent interest. Among the results we prove are the theorem that the fundamental group of a separable, connected, locally path connected, one-dimensional metric space is free if and only if it is countable if and only if the space has a universal cover and the theorem that the fundamental group of a compact, one-dimensional, connected metric space embeds in an inverse limit of finitely generated free groups and is shape injective.     

On stable cohomotopy groups of compact spaces

The stable cohomotopy theory is a useful tool for studying global properties of compact spaces. In the stable shape theory we have infinite-dimensional versions of the Whitehead theorem. They require to use the stable cohomotopy groups.     

On modal logics arising from scattered locally compact Hausdorff spaces

For a topological space X, let $\mathsf{L}(X)$ be the modal logic of X where □ is interpreted as interior (and hence ◇ as closure) in X. It was shown in [3] that the modal logics S4, S4.1, S4.2, S4.1.2, S4.Grz, ${\mathsf{S4}.\mathsf{Grz}}_n$ ($n\ge 1$), and their intersections arise as $\mathsf{L}(X)$ for some Stone space X. We give an example of a scattered Stone space whose logic is not such an intersection. This gives an affirmative answer to [3, Question 6.2]. On the other hand, we show that a scattered Stone space that is in addition hereditarily paracompact does not give rise to a new logic; namely we show that the logic of such a space is either S4.Grz or ${\mathsf{S4}.\mathsf{Grz}}_n$ for some $n\ge 1$. In fact, we prove this result for any scattered locally compact open hereditarily collectionwise normal and open hereditarily strongly zero-dimensional space.     

Hausdorff operators on compact abelian groups

Necessary and sufficient conditions are given for the boundedness of Hausdorff operators on the generalized Hardy spaces $H_E^p(G)$, real Hardy space $H_{\mathbb{R}}^1(G)$, $\text{BMO}(G)$, and $\text{BMOA}(G)$ for compact Abelian group G. Surprisingly, these conditions turned out to be the same for all groups and spaces under consideration. Applications to Dirichlet series are given. The case of the space of continuous functions on G and examples are also considered.     

On the Hausdorff and packing measures of typical compact metric spaces

We study the Hausdorff and packing measures of typical compact metric spaces belonging to the Gromov–Hausdorff space (of all compact metric spaces) equipped with the Gromov–Hausdorff metric.     

Not all compact Hausdorff spaces are supercompact

De Groot and Verbeek have both asked for an example of a compact Hausdorff space which is not supercompact. Is is shown here that if X is not pseudocompact, then βX is not supercompact. It is done in the more general setting of Wallman compactifications.     

Noncommutative recurrence over locally compact Hausdorff groups

We extend previous results on noncommutative recurrence in unital *-algebras over the integers to the case where one works over locally compact Hausdorff groups. We derive a generalization of Khintchine's recurrence theorem, as well as a form of multiple recurrence. This is done using the mean ergodic theorem in Hilbert space, via the GNS construction.     

On quotient spaces of compact lie groups by Tori centralizers

We consider the graph of the homogeneous space K/L, where K is a compact Lie group and L is the centralizer of a torus in K. We obtain a characterization of those spaces whose graphs admit embeddings in a certain standard graph. We compute the number of arcs in such graphs. We also give a simple expression for the Euler class of the homogeneous space K/L.     

On the role of finite,hereditarily normal spaces and maps in the genesis of compact Hausdorff spaces

We consider how properties of the bonding maps of the inverse spectrum determine properties of the inverse limit. Specifically, we study the limits of inverse spectra of finite T₀-spaces with bonding maps which are either chaining or normalizing. We will show that if the bonding maps are normalizing, then the inverse limit is a normal T₀-space, and therefore, its Hausdorff reflection is its subset of specialization minimal elements. If the maps are chaining, then the inverse limit is a completely normal spectral space; such spaces have been studied since they include the real spectra of commutative rings [C.N. Delzell, J.J. Madden, J. Algebra 169 (1994) 71], and the prime spectrum of a ring of functions, Spec(C(X)). The existence and importance of this class of non-Hausdorff, normal topological spaces was extremely surprising to us. Further, each of these results is reversible; if the inverse limit is normal, then each space in the spectrum is preceded by one whose bonding map to it is normalizing. By way of contrast, the inverse limit of finite T₀-spaces with separating bonding maps need not be a normal topological space (Example 3.8(a)) and furthermore, if the spaces of the inverse spectrum are normal, then the Hausdorff reflection of the limit must be zero-dimensional (Theorem 3.15).     

Lipschitz spaces on compact Lie groups

We prove the equivalence between Lipschitz spaces on a compact Lie group defined in terms of Weierstrass integrals and by means of higher order difference operators.     

Countable compact Hausdorff spaces need not be metrizable in ZF

We show that the existence of a countable, first countable, zero-dimensional, compact Hausdorff space which is not second countable, hence not metrizable, is consistent with ZF.