Rim-compact Spaces Research Articles

Topologizing homeomorphism groups of rim-compact spaces

Let X be a Tychonoff space, H ( X ) the group of all self-homeomorphisms of X with the usual composition and e : ( f , x ) ∈ H ( X ) × X → f ( x ) ∈ X the evaluation function. Topologies on H ( X ) providing continuity of the evaluation function are called admissible. Topologies on H ( X ) compatible with the group operations are called group topologies. Whenever X is locally compact T 2 , there is the minimum among all admissible group topologies on H ( X ) . That can be described simply as a set-open topology, further agreeing with the compact-open topology if X is also locally connected. We show the same result in two essentially different cases of rim-compactness. The former one, where X is rim-compact T 2 and locally connected. The latter one, where X agrees with the rational number space Q equipped with the euclidean topology. In the first case the minimal admissible group topology on H ( X ) is the closed-open topology determined by all closed sets with compact boundaries contained in some component of X. Moreover, whenever X is also separable metric, it is Polish. In the rational case the minimal admissible group topology on H ( Q ) is just the closed-open topology. In both cases the minimal admissible group topology on H ( X ) is closely linked to the Freudenthal compactification of X. The Freudenthal compactification in rim-compactness plays a key role as the one-point compactification does in local compactness. In the rational case we investigate whether the fine or Whitney topology on H ( Q ) induces an admissible group topology on H ( Q ) stronger than the closed-open topology.

A generating family for the Freudenthal compactification of a class of rimcompact spaces

A generating family for the Freudenthal compactification of a class of rimcompact spaces

The dimension of remainders of rim-compact spaces

Answering a question of Isbell we show that there exists a rim-compact space X such that every compactification Y of X has dim(Y \X) ≥ 1. It is known that a space X is rim-compact if and only if X has a compactification Y such that Y X is zero-dimensionally embedded in Y , i.e. Y has a base for the open sets whose boundaries are disjoint from Y X. Note that this implies that ind(Y \X) ≤ 0. On the other hand, if there is a compactification Y of the space X such that ind(Y \X) ≤ 0 then X need not be rim-compact; an example was given by Smirnov in [1958]. If X is Lindelof at infinity, i.e. each compact subset of X is contained in a compact set with a countable base for its neighbourhoods, then X is rimcompact if and only if it has a compactification Y such that ind(Y \X) ≤ 0; such a compactification then also has dim(Y X) ≤ 0. This suggests the question whether every rim-compact space X has a compactification Y such that dim(Y \X) ≤ 0 (see Isbell [1964] and Aarts and Nishiura [1993]; see also Diamond, Hatzenbuhler and Mattson [1988] for related problems). We shall show that the answer is negative. We construct a strongly zero-dimensional spaceX such that for every compactification Y ofX we have dim(Y \X) ≥ 1. Our space X will be such that βX \X is metrizable, zero-dimensional but not strongly zero-dimensional. Originally, the remainder was Roy’s space ∆; however, by using Kulesza’s space we were able to obtain an example of the smallest possible weight ω1. Convention. We identify an ordinal with its set of predecessors so that for example ω1 + 1 = ω1 ∪ {ω1}. All ordinals under consideration carry the order topology. 1991 Mathematics Subject Classification: Primary 54F45; Secondary 54D35. 288 J. M. Aarts and E. Coplakova The construction. Let K be Kulesza’s space, i.e. K is a (completely) metrizable subspace of ω 1 of weight w(K) = ω1 which is dense in ω ω 1 and satisfies indK = 0 and IndK = 1 (see Kulesza [1990]). Consider now Z = (ω1 + 1) = (ω1 + 1) × (ω1 + 1) and let X = Z (K × {ω1}) . Then (ω1+1)×ω1 ⊆ X ⊆ Z. As (ω1+1) is compact we may conclude that Z = (ω1 +1)×(ω1 +1) is the Cech–Stone compactification of (ω1 +1)×ω1 (see Engelking [1989, Problem 3.12.20(c)]). Hence βX = Z as well. Note that βX is a product of compact zero-dimensional spaces hence βX is also compact and zero-dimensional and therefore strongly zero-dimensional. It follows that X itself is strongly zero-dimensional, hence zero-dimensional and a fortiori rim-compact. It is also easily seen that w(X) = ω1. It remains to show that dim(αX \X) ≥ 1 for every compactification αX of X. Let αX be a compactification of X. Consider f :βX → αX, the extension of the natural embedding idX :X → αX over βX. As βX and αX are compact, the mapping f is perfect. Now, βX \X = f−1[αX \X] so f βX \X is also perfect. But βX \X = K × {ω1} and K is metrizable. To finish our argument we need the following theorem, due to Morita and Nagami (see Engelking [1989]). Theorem. If f :X → Y is a closed mapping of a metrizable space X to a metrizable space Y and for every y ∈ Y , Ind f−1[{y}] ≤ k for k ≥ 0, then IndX ≤ IndY + k. Now, K×{ω1} and αX \X are metrizable, the mapping f :K×{ω1} → αX \X is perfect and for each x ∈ αX \X the fiber f−1[{x}] is compact and zero-dimensional, hence strongly zero-dimensional. Note that this means that Ind(f−1[{x}]) ≤ 0 for each x ∈ αX \X, so by the theorem Ind(K × {ω1}) ≤ Ind(αX \X) + 0 , hence dim(αX \X) = Ind(αX \X) ≥ Ind(K × {ω1}) = 1 .

Wallman prebases

Let X be completely regular and Hausdorff. We define the notion of a Wallman prebase for X in a manner analogous to that of the Wallman base used in the construction of a Wallman compactification of X. Corresponding to each Wallman prebase of X is a closed completely regular Hausdorff image of X. Conversely, every closed completely regualr Hausdorff image Y of X gives rise to a number of Wallman prebases of X, one for each Wallman base of Y. A natural way of identifying “equivalent” Wallman prebases yields a 1-1 correspondence between prebases and closed images. Restrictions on the Wallman prebase yield images and maps satisfying stronger conditions. In particular, a space is the perfect preimage of a rimcompact space if and only if it has a Wallman prebase satisfying certain conditions.

Almost rimcompact spaces

A 0- space is a completely regular Hausdorff space possesing a compactification with zero-dimensional remainder. Recall that a space X is called rimcompact if X has a basis of open sets with compact boundaries. It is well known that X is rimcompact if and only if X has a compactification which has a basis of open sets whose boundaries are contained in X. Thus any rimcompact space is a 0-space; the converse is not true. In this paper the class of almost rimcompact spaces is introduced and shown to be intermediate between the classes of rimcompact spaces and 0-spaces. It is shown that a space X is almost rimcompact if and only if X has a compactification in which each point of the remainder has a basis (in the compactification) of open sets whose boundaries are contained in X.

Paracompactness in normal, locally connected, rim-compact spaces

We prove that a locally connected, rim-Lindelöf, submeta Lindelöf, normal, ⩽ω 2-collectionwise T 2 space is paracompact. Further, if ‘Lindelöf’ is replaced by ‘compact’, then ‘⩽ω 2-collectionwise T 2’ can be omitted.

Some properties of almost rimcompact spaces

Some properties of almost rimcompact spaces

Regular Wallman compactifications of rim-compact spaces

AbstractA compact Hausdorff space is regular Wallman if it possesses a separating ring of regular closed sets, an s-ring. It was proved by P. C. Baayen and J. van Mill [General Topology and Appl. 9 (1978), 125–129] that if a locally compact Hausdorff space possesses an s-ring, then every Hausdorff compactification with zero-dimensional remainder is regular Wallman.In this paper the reasoning leading to this result is modified to work in a more general setting. Iet αX be a Hausdorff compactification of a space X, and let be the family of those closed sets in αX whose boundaries are contained in X. A main result is the following: If contains an s-ring for some Hausdorff compactification γX, then every larger Hausdorff compactification αX for which is a base for the closed sets on αX — X, is regular Wallman. Various consequences concerning compactifications of a class of rim-compact spaces (called totally rim-compact spaces) are discussed.

Concerning σ-Connectedness of Baire Spaces

A well known theorem of Sierpiński states that every compact connected Hausdorff space is σ-connected. Hence, if X is locally compact and Hausdorff and X is locally connected at x, then x has a σ-connected neighborhood. However, local connectedness at x is not a necessary condition for x to have a σ-connected neighborhood, because the whole space may be σ-connected without being locally connected at x. One of the purposes of the present paper is then to investigate which points of a given locally compact Hausdorff space have σ-connected neighborhoods. We find also sufficient conditions for a connected, hereditarily Baire space to be σ-connected and prove the impossibility of expressing a connected, Čech-complete, rim compact space as a countable infinite union of mutually disjoint compact sets. Finally, we introduce the concept of D-connected space and relate it to σ-connectedness.

On Levine′s Decomposition of Continuity

AbstractA strong version of Levine′s decomposition of continuity leads to the result that a closed graph weakly continuous function into a rim-compact space is continuous. This result implies a closed graph theorem: every almost continuous closed graph function into a strongly locally compact space is continuous. An open problem of Shwu-Yeng T. Lin and Y.-F. Lin asks if every almost continuous closed graph function from a Baire space to a second countable space is necessarily continuous. This question is answered in the negative by an example.

A maximal realcompactification with 0-dimensional outgrowth

If X X is a rimhard space, then it has a realcompactification with properties similar to those of the Freudenthal compactification of a rimcompact space.

On Fan-Gottesman compactification and P. S. Alexsandrov's problems

As a generalization of Freudenthal’s compactification of rim-compact Hausdorff spaces [S], Fan and Gottesman introduced the concept of normal base and have constructed a compactification for any regular space with a normal base [S]. The Fan-Gottesman compactification includes Wallman compactification of normal spaces and Aleksandrov one-point compactification of locally compact spaces as particular cases. The results in the present paper are mainly consequences of Fan-Gottesman Theorem [5, Theorem 11; other concepts employed are Aleksandrov compactification ciT of a completely regular space T [I], [7], W 11 a man H-closed extension w(X) of a Hausdorff space X, and the maximal binding families relative to all open sets in a space [5]. A new concept called pseudonormality, analogous to but weaker than normality, is introduced. A regular pseudonormal space is completely regular but the converse is untrue. There are pseudonormal spaces that are not normal, namely, the well-known Sorgenfrey plank, Tychonoff plank, and the Moore (Niemytzki) plane. As a real-valued continuous function f (bounded or unbounded) on a Hausdorff space X can be continuously extended over w(X), denoted by f, we will show that f has a unique extension [ on the collection X** of maximal binding families in X in some sense described below. A Hausdorff space X is weakly periphery-compact if and only if continuous functions j on w(X) separate points in w(X); and a regular space X is pseudonormal if and only if all Jon X* * separate points in X**. A Hausdorff space is completely regular if and only if for each pair (x, y), x E X, y E X**, and x # y, there is a continuous function f on X such that f(x) #J(y). A new construction for the Stone-Tech compactification j3T of a completely regular space T, based on the Wallman H-closed extension W(T), will be given and hence a new proof for the Stone-Tech extension theorem. The

Ends of locally compact groups and their coset spaces

Freudenthal [5, 7] defined a compactification of a rim-compact space, that is, a space having a base of open sets with compact boundary. The additional points are called ends and Freudenthal showed that a connected locally compact non-compact group having a countable base has one or two ends. Later, Freudenthal [8], Zippin [16], and Iwasawa [11] showed that a connected locally compact group has two ends if and only if it is the direct product of a compact group and the reals.

On Homeomorphisms between Extension Spaces

In this note, conditions are obtained which will ensure that two topological spaces are homeomorphic when they have homeomorphic extension spaces of a certain kind. To discuss this topic in suitably general terms, an unspecified extension procedure, assumed to be applicable to some class of topological spaces, is considered first, and it is shown that simple conditions imposed on the extension procedure and its domain of operation easily lead to a condition of the desired kind. After the general result has been established it is shown to be applicable to a number of particular extensions, such as the Stone-Čech compactification and the Hewitt Q-extension of a completely regular Hausdorff space, Katětov's maximal Hausdorff-closed extension of a Hausdorff space, the maximal zero-dimensional compactification of a zero-dimensional space, the maximal Hausdorff-minimal extension of a semi-regular space, and Freudenthal's compactification of a rim-compact space. The case of the Hewitt Q-extension was first discussed by Heider (6).