Almost arcwise connected continua without dense arc components

A continuum M is almost arcwise connected if each pair of nonempty open subsets of M can be joined by an arc in M. An almost arcwise connected plane continuum without a dense arc component can be defined by identifying pairs of endpoints of three copies of the Knaster indecomposable continuum that has two endpoints. In [7] K.R. Kellum gave this example and asked if every almost arcwise connected continuum without a dense arc component has uncountably many arc components. We answer Kellum's question by defining an almost arcwise connected plane continuum with only three arc components none of which are dense. A continuum M is almost Peano if for each finite collection $\text{C}$ of nonempty open subsets of M there is a Peano continuum in M that intersects each element of $\text{C}$. We define a hereditarily unicoherent almost Peano plane continuum that does not have a dense arc component. We prove that every almost arcwise connected planar λ-dendroid has exactly one dense arc component. It follows that every hereditarily unicoherent almost arcwise connected plane continuum without a dense arc component has uncountably many arc components. Using an example of J. Krasinkiewicz and P Minc [8], we define an almost Peano λ-dendroid that do not have a dense arc component. Using a theorem of J.B. Fugate and L. Mohler [3], we prove that every almost arcwise connected λ-dendroid without a dense arc component has uncountably many arc components. In Euclidean 3-space we define an almost Peano continuum with only countably many arc components no one of which is dense. It is not known if the plane contains a continuum with these properties.     

Remainders of arcwise connected compactifications of the plane

A. Lelek asked which continua are remainders of locally connected compactifications of the plane. In this paper we study a similar problem with local connectedness replaced by arcwise connectedness. (Each locally connected continuum is arcwise connected.) We give the following characterization: a continuum X is pointed 1-movable if and only if there is an arcwise connected compactification of the plane with X as the remainder.     

Whitney preserving maps on finite graphs

For a Whitney preserving map f:X→G we show the following: (a) If X is arcwise connected and G is a graph which is not a simple closed curve, then f is a homeomorphism; (b) If X is locally connected and G is a simple closed curve, then X is homeomorphic to either the unit interval [0,1], or the unit circle S¹. As a consequence of these results, we characterize all Whitney preserving maps between finite graphs. We also show that every hereditarily weakly confluent Whitney preserving map between locally connected continua is a homeomorphism.     

Various types of local connectedness in n-fold hyperspaces

Let X be a continuum. The n-fold hyperspace Cn(X), n<∞, is the space of all nonempty compact subsets of X with the Hausdorff metric. Four types of local connectivity at points of Cn(X) are investigated: connected im kleinen, locally connected, arcwise connected im kleinen and locally arcwise connected. Characterizations, as well as necessary or sufficient conditions, are obtained for Cn(X) to have one or another of the local connectivity properties at a given point. Several results involve the property of Kelley or C^*-smoothness. Some new results are obtained for C(X), the space of subcontinua of X. A class of continua X is given for which Cn(X) is connected im kleinen only at subcontinua of X and for which any two such subcontinua must intersect.     

Some generalizations of the concept of a p-space

In this article, some generalizations of the concept of a p-space are introduced and studied. The notion of a source of a space in a larger space and the concepts of partial plumage, s-embedding, p-embedding, p^?-embedding, s-space, and p^?-space are defined and studied in depth (see Theorems 2.6, 2.7, 3.2, 4.3, 4.4, 4.10 and their corollaries). An example of a hereditarily p^?-space which is not a p-space and is a perfect image of a hereditarily p-space is indicated (Example 2.9). Among the main results, we establish that if a paracompact space X is p-embedded in a pseudocompact space as a dense subspace, then X is a p-space (Corollary 4.8), and that if X has a countable network and is p^?-embedded in a pseudocompact space, then X is metrizable (Corollary 4.11). The following problem is posed: is every paracompact Gδ-subspace of a pseudocompact space ?ech-complete?     

Open selections on smooth fans

Given a dendroid X, an open selection is an open map such that s(A)∈A for every A∈C(X). We show that a smooth fan X admits an open selection if and only if X is locally connected.     

Domain representability of certain function spaces

Let Cp(X) be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that Cp(X) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then Cp(X) is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space Cp(X), Oxtoby's pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement “every countable subset of X is closed”.     

Hereditarily indecomposable subcontinua of the product of two Souslin arcs are metric

We prove that if each of X and Y is a Souslin arc (a Hausdorff arc that is the compactification of a connected Souslin line), then every hereditarily indecomposable subcontinuum of X×Y is metric.     

Lifting paths on quotient spaces

Let X be a compactum and G an upper semi-continuous decomposition of X such that each element of G is the continuous image of an ordered compactum. If the quotient space X/G is the continuous image of an ordered compactum, under what conditions is X also the continuous image of an ordered compactum? Examples around the (non-metric) Hahn-Mazurkiewicz Theorem show that one must place severe conditions on G if one wishes to obtain positive results. We prove that the compactum X is the image of an ordered compactum when each g∈G has 0-dimensional boundary. We also consider the case when G has only countably many non-degenerate elements. These results extend earlier work of the first named author in a number of ways.     

Strongly e-movable convergence and spaces of ANR's

Let X be a metric space and let ANR(X) denote the hyperspace of all compact ANR's in X. This paper introduces a notion of a strongly e-movable convergence for sequences in ANR(X) and proves several characterizations of strongly e-movable convergence. For a (complete) separable metric space X we show that ANR(X) with the topology induced by strongly e-movable convergence can be metrized as a (complete) separable metric space. Moreover, if X is a finite-dimensional compactum, then strongly e-movable convergence induces on ANR(X) the same topology as that induced by Borsuk's homotopy metric.For a separable Q-manifold M, ANR(M) is locally arcwise connected and A, B ? ANR(M) can be joined by an arc in ANR(M) iff there is a simple homotopy equivalence ?: A → B homotopic to the inclusion of A into M.     

On an internal property of absolute retracts,II

Let X be a (metrizable) space. A mixer for X is, roughly speaking, a map μ:X³→X such that μ(x, x, y) = μ(x, y, x) = μ(y, x, x) = x for all x, y ∈ X. We show that each AR has a mixer and that a finite dimensional path connected space with a mixer is an AR. Our main result is that each separable space with a mixer and having an open cover by sets contractible within the whole space, is LEC.     

On nagata's problem for paracompact σ-metric spaces

Let (A) be the characterization of dimension as follows: Ind X?n if and only if X has a σ-closure-preserving base $\text{W}$ such that Ind B(W)?n?1 for every W?$\text{W}$. The validity of (A) is proved for spaces X such that(i) X is a paracompact σ-metric space with a scale {X_i} such that each X_i has a uniformly approaching anti-cover, or(ii) X is a subspace of the product ΠX_i of countably many L-spaces X_i, the notion of which is due to K. Nagami.(i) and (ii) are the partial answers to Nagata's problem wheter (A) holds or not for every M₁-space X.     

On sieve-complete and compact-like spaces

It is shown that a completely regular space X is sieve-complete (or, equivalenty, X is the open image of a paracompact ?ech-complete space) iff βX?X is compact-like, i.e., Player I has a winning strategy in the topological game G(C, βX?X) of [13].     

A study of cut points in connected topological spaces

We prove that every H(i) subset H of a connected space X such that there is no proper connected subset of X containing H, contains at least two non-cut points of X. This is used to prove that a connected space X is a COTS with endpoints if and only if X has at most two non-cut points and has an H(i) subset H such that there is no proper connected subset of X containing H. Also we obtain some other characterizations of COTS with endpoints and some characterizations of the closed unit interval.     

Transitive points via Furstenberg family

Let (X,T) be a topological dynamical system and F be a Furstenberg family (a collection of subsets of Z₊ with hereditary upward property). A point x∈X is called an F-transitive one if {n∈Z₊:Tnx∈U}∈F for every non-empty open subset U of X; the system (X,T) is called F-point transitive if there exists some F-transitive point. In this paper, we aim to classify transitive systems by F-point transitivity. Among other things, it is shown that (X,T) is a weakly mixing E-system (resp. weakly mixing M-system, HY-system) if and only if it is {D-sets}-point transitive (resp. {central sets}-point transitive, {weakly thick sets}-point transitive).It is shown that every weakly mixing system is Fip-point transitive, while we construct an Fip-point transitive system which is not weakly mixing. As applications, we show that every transitive system with dense small periodic sets is disjoint from every totally minimal system and a system is Δ^?(Fwt)-transitive if and only if it is weakly disjoint from every P-system.     

The D-property of some Lindelöf spaces and related conclusions

It is shown that the space Cp(τω) is a D-space for any ordinal number τ, where . This conclusion gives a positive answer to R.Z. Buzyakova's question. We also prove that another special example of Lindelöf space is a D-space. We discuss the D-property of spaces with point-countable weak bases. We prove that if a space X has a point-countable weak base, then X is a D-space. By this conclusion and one of T. Hoshina's conclusion, we have that if X is a countably compact space with a point-countable weak base, then X is a compact metrizable space. In the last part, we show that if a space X is a finite union of θ-refinable spaces, then X is a αD-space.     

A glance at spaces with closure-preserving local bases

Call a space X (weakly) Japanese at a pointx∈X if X has a closure-preserving local base (or quasi-base respectively) at the point x. The space X is (weakly) Japanese if it is (weakly) Japanese at every x∈X. We prove, in particular, that any generalized ordered space is Japanese and that the property of being (weakly) Japanese is preserved by σ-products; besides, a dyadic compact space is weakly Japanese if and only if it is metrizable. It turns out that every scattered Corson compact space is Japanese while there exist even Eberlein compact spaces which are not weakly Japanese. We show that a continuous image of a compact first countable space can fail to be weakly Japanese so the (weak) Japanese property is not preserved by perfect maps. Another interesting property of Japanese spaces is their tightness-monolithity, i.e., in every weakly Japanese space X we have for any set A⊂X.     

Hereditarily indecomposable subcontinua of inverse limits of Souslin arcs are metric

We prove that if Si is a Souslin arc (a Hausdorff arc that is the compactification of a Souslin line) for each i and , then every hereditarily indecomposable subcontinuum of X is metric. Since every non-degenerate hereditarily indecomposable continuum that is an inverse limit on metric arcs is a pseudo-arc, it follows that such an X would be a pseudo-arc or a point.     

On metric spaces with the Haver property which are Menger spaces

A metric space (X,d) has the Haver property if for each sequence ?₁,?₂,… of positive numbers there exist disjoint open collections V₁,V₂,… of open subsets of X, with diameters of members of Vi less than ?i and covering X, and the Menger property is a classical covering counterpart to σ-compactness. We show that, under Martin's Axiom MA, the metric square (X,d)×(X,d) of a separable metric space with the Haver property can fail this property, even if X² is a Menger space, and that there is a separable normed linear Menger space M such that (M,d) has the Haver property for every translation invariant metric d generating the topology of M, but not for every metric generating the topology. These results answer some questions by L. Babinkostova [L. Babinkostova, When does the Haver property imply selective screenability? Topology Appl. 154 (2007) 1971-1979; L. Babinkostova, Selective screenability in topological groups, Topology Appl. 156 (1) (2008) 2-9].     

On metric spaces with the properties of de Groot and Nagata in dimension one

A metric space (X,d) has the de Groot property GPn if for any points x₀,x₁,…,xn+2∈X there are positive indices i,j,k?n+2 such that i≠j and d(xi,xj)?d(x₀,xk). If, in addition, k∈{i,j} then X is said to have the Nagata property NPn. It is known that a compact metrizable space X has dimension dim(X)?n iff X has an admissible GPn-metric iff X has an admissible NPn-metric.We prove that an embedding f:(0,1)→X of the interval (0,1)⊂R into a locally connected metric space X with property GP₁ (resp. NP₁) is open, provided f is an isometric embedding (resp. f has distortion Dist(f)=‖fLip_‖⋅‖f−1Lip_‖<2). This implies that the Euclidean metric cannot be extended from the interval [−1,1] to an admissible GP₁-metric on the triode T=[−1,1]∪[0,i]. Another corollary says that a topologically homogeneous GP₁-space cannot contain an isometric copy of the interval (0,1) and a topological copy of the triode T simultaneously. Also we prove that a GP₁-metric space X containing an isometric copy of each compact NP₁-metric space has density ?c.