A topological space is said to be ω-bounded if each of its countable subsets has compact closure. It has been shown recently by Itzkowitz and Shakhmatov that for every compact Abelian group G of uncountable weight, and for every compact connected group of G of uncountable weight, the set Ω(G) of dense ω-bounded subgroups of G satisfies Ω(G) ⩾ G. These authors asked whether 0977 their estimate Ω(G) ⩾ G may be improved to Ω(G) = 2G for some or all such G. In the 0977 0605 V 3 present paper we answer this question affirmatively for all compact groups G which are either Abelian or connected and which satisfy in addition the condition w(G) = (w(G))ω. We show also that every compact group G with ω(G) ⩾ log((2′)+) satisfies Ω(G) > 2′.

We show that there exist Canter sets in the circle that are not extendable to sets that meet every line in the plane in exactly two points. This result solves a problem that was formulated by R. D. Mauldin.

We provide a simpler proof of Gouweleeuw's theorem about the convexity of the range of an Rn-valued vector measure μ in terms of μ. We also discuss possible extensions of Gouweleeuw's results to vector measures with values in infinite-dimensional vector spaces and to unbounded vector measures.

Every space is assumed to be separable and metric. A space is called (strongly) countably dimensional if it can be written as a countable union of (closed) finite dimensional subspaces. A space X is called strongly infinite dimensional if the space admits an essential system (Fn, Gn) ∞ n=1, i.e. Fn and Gn are disjoint closed subsets of X such that if Sn is a closed separator of Fn and Gn for each n, then ⋂∞ n=1 Sn is nonempty. The sequence of left and right endfaces of the Hilbert cube is the standard example of an essential system. A well-known theorem of Engelking [E] states that every autohomeomorphism h of an n-dimensional space X can be extended to a homeomorphism h : C → C, where C is an n-dimensional compactification of X (and hence we have a ≤ndimensional remainder). We consider the question of whether similar results can be obtained for infinite dimensional spaces, i.e. is it possible to put a bound on the dimension of the remainder? The following example shows that the answer is no if we allow incomplete spaces. Consider the Hilbert cube Q = [0, 1] and the strongly countably dimensional pseudoboundary σ = {x ∈ Q : xi = 0 from some index on}. It was shown by R. D. Anderson that Q σ is homeomorphic to Hilbert space (see [BP, Theorem V.5.1]). The following proposition is a slight improvement of the known result that the remainder of every compactification of σ contains a copy of Q.

We determine the topological structure of the subsets of the hyperspace 2(R2) consisting of absolute retracts and of absolute neighborhood retracts respectively.

Let X be a separable and metrizable space containing uncountably many pairwise disjoint copies of the compactum K. We discuss the question whether X must contain K x 2(omega).

We prove that every compact Basically Disconnected space of π \pi -weight ω 1 {\omega _1} has a dense Extremally Disconnected subspace. In Boolean algebraic terms: every σ \sigma -complete Boolean algebra B with density ω 1 {\omega _1} carries an ultrafilter which generates an ultrafilter in the completion of B. The statement that every compact Basically Disconnected space of weight c \mathfrak {c} has a dense Extremally Disconnected subspace is shown to be equivalent to CH.

Adapting terminology suggested by work of E. Hewitt [Duke Math. J. 10 (1943), 309-333], we say that a group G G is strongly resolvable if for every nondiscrete Hausdorff group topology I \mathcal {I} on G G there is D ⊆ G D \subseteq G such that both D D and G ∖ D G\backslash D are I \mathcal {I} -dense in G G . Theorem. Let G G be an Abelian group. (a) If G G contains no subgroup isomorphic to the group ⨁ ω { 0.1 } { \bigoplus _\omega }\{ 0.1\} , then G G is strongly resolvable. (b) Assume MA. If G G contains a copy of ⨁ ω { 0 , 1 } { \bigoplus _\omega }\{ 0,1\} , then G G is not strongly resolvable. Our proof of (b) depends heavily on work of Malykhin.

If X is a space then L(X) denotes the subspace of C(X) consisting of all Peano (sub)continua. We prove that for n greater-than-or-equal-to 3 the space L(R(n)) is homeomorphic to B(infinity), where B denotes the pseudo-boundary of the Hilbert cube Q.

CH implies that every homeomorphism of nowhere dense closed P-sets in beta omega\omega can be extended to an autohomeomorphism of beta omega\omega. CH also implies that every closed P-set of weight at most c of a compact zero-dimensional F-space is a retract.

If X is an infinite product of non-degenerate Peano continua then the set dim~(X) = {A~2x:dim A = ~) is an F,,,-absorber in 2'. As a consequence, there is a homeomorphism f: 2x --. 6°° such that f(dim~(X)) = B', where B denotes the pseudo-boundary of the Hilbert cube Q. There is a locally infinite-dimensional Peano continuum X such that for every n, dim~(Xn) is not homeomorphic to BOO.

The aim of this paper is to define absorbing systems in infinite-dimensional manifolds and to derive some basic properties of them along the lines of Chapman. As an application we prove that for a countable nondiscrete Tychonov space X, if Cp(X) is and Fσδ subset of RX then it is an Fσδ-absorber, and hence homeomorphic to the countable infinite product of copies of l2ƒ. This generalizes a result of Dobrowolski, Marciszewski and Mogilski.

This is a cumulative status report on the 1100 problems listed in the volume Open Problems in Topology (North-Holland, 1990), edited by J. van Mill and G.M. Reed.

Sierpiński invented in 1932 a method for constructing examples, which is now known as the “technique of killing homeomorphisms”. This method was used by Ohkuma to construct a rigid homogeneous chain, and by van Douwen to construct a compact homogeneous space with a measure that “knows” which sets are homeomorphic, and by Keesling and Wilson to construct an almost uniquely homogeneous subgroup of Rn. The aim of this paper is to derive these results simultaneously.

We construct an example of an AR-map f : X → Y f:X \to Y , where X X is a strongly countable dimensional compact AR and Y Y is a countable dimensional AR which is not strongly countable dimensional. Using this map we find a shrinkable decomposition of the pre-Hilbert space l f 2 l_f^2 whose quotient map does not stabilize to a near homeomorphism. We also present a partial result concerning the question whether cell-like maps preserve countable dimensionality.

Let n n and k k be fixed integers such that n ≥ 1 n \geq 1 and 0 ≤ k ≤ n 0 \leq k \leq n . Let B k n B_k^n and s k n s_k^n denote the k k -dimensional universal pseudo-boundary and the k k -dimensional universal pseudo-interior in R n {{\mathbf {R}}^n} , respectively. The aim of this paper is to prove that B k n B_k^n is homeomorphic to B k m B_k^m if and only if s k n s_k^n is homeomorphic to s k m s_k^m if and only if n = m n = m or n n , m ≥ 2 k + 1 m \geq 2k + 1 .

We show that there exists a homeomorphism from the hyperspace of the Hilbert cube Q onto the countable product of Hilbert cubes such that the greater-than-or-equal-to k-dimensional sets are mapped onto B(k) x Q x Q x ..., where B is the pseudoboundary of Q. In particular, the infinite-dimensional compacta are mapped onto B(omega), which is homeomorphic to the countably infinite product of l(f)2. In addition, we prove for k is-an-element-of {1, 2, ... , infinity} that the space of uniformity greater-than-or-equal-to k-dimensional sets in 2Q is also homeomorphic to (l(f)2)omega.

Continuing earlier investigations into the question of the existence of a proper dense subgroup of a given topological group, the authors obtain some positive and some negative results, as follows: (a) Every non-degenerate connected Abelian group contains a proper dense subgroup. (b) Every infinite pseudocompact group contains a proper dense subgroup. (c) If G is a totally bounded Abelian group with wG = \G\ = alpha > omega, and if in addition G is torsion-free (or more generally, if \G\ tor \G\ = alpha), then G has a proper dense subgroup. (d) For every strong limit cardinal-alpha such that cf(alpha) = omega there is a totally bounded Abelian torsion group G such that wG = \G\ = alpha and G has no proper dense subgroup.Among the questions left unsettled is this: Can the conditions in (d) on alpha be relaxed or even omitted?

The space under consideration is the basic fake Hilbert space Y of Anderson, Curtis and van Mill. It is shown that the product of an arbitrary space A with Y is homeomorphic to Y if and only if A is a compact absolute retract. Furthermore, we prove that the complement of Y x Y is a capset in Q x Q, which implies the known result that Y x Y is homeomorphic to Hilbert space.