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It is proved in ZF that if a collection of (quasi)-metric spaces is indexed by a countable union of finite sets, then the product of this collection is (quasi)-metrizable, while it is independent of ZF that every countable product of metrizable spaces is quasi-metrizable. It is also proved that if J is a non-empty set, while a space X, consisting of at least two points, is equipped with the co-finite topology, then the product X J is quasi-metrizable if and only if both X and J are countable unions of finite sets. Several equivalences of CUT ( fin ) are deduced. It is shown that, in every model of ZF + ¬ CUT ( fin ) , for an uncountable set J, the space N J can be normal (even metrizable), a Cantor cube can be simultaneously metrizable, not second-countable and non-compact.

Let X and Y be metric continua. We consider the following property (*): if M is a subcontinuum of X × Y such that π X ( M ) = X and π Y ( M ) = Y , where π X and π Y are the respective projections on X and Y, then M has small connected neighborhoods in X × Y . Property (*) has been studied by D. P. Bellamy, J. M. Łysko and the first named author. In this paper we continue studying property (*) in products of continua. We prove: (a) the product of homogeneous continua having the fixed point property has property (*); (b) the product of a solenoid and any Knaster continuum has property (*); (c) there exists a Kelley continuum X such that X × [ 0 , 1 ] does not have property (*); and (d) the product of a chainable Kelley continuum and [ 0 , 1 ] has property (*).

The paper is devoted to study the $n$th Hawaiian group $\mathcal{H}_n$, $n \ge 1$, of the wedge sum of two spaces $(X,x_*) = (X_1, x_1) \vee (X_2, x_2)$. Indeed, we are going to give some versions of the van Kampen theorem for Hawaiian groups of the wedge sum of spaces. First, among some results on Hawaiian groups of semilocally strongly contractible spaces, we present a structure for the $n$th Hawaiian group of the wedge sum of CW-complexes. Second, we give more informative structures for the $n$th Hawaiian group of the wedge sum $X$, when $X$ is semilocally $n$-simply connected at $x_*$. Finally, as a consequence, by generalizing the well-known Griffiths space for dimension $n\geq 1$, we give some information about the structure of Hawaiian groups of Griffiths spaces at any points., Comment: 11 pages

We prove the analog of Pasynkov's result on finite-dimensionality of topological products for the dimension-like function I defined by S. Iliadis. Let F j be a normal base on a topological space X j , j = 1 , 2 . Then I ( X 1 × X 2 , F 1 ⊗ F 2 ) ≤ φ ( I ( X 1 , F 1 ) , I ( X 2 , F 2 ) ) , where φ is a recursion relation. As a consequence we get the result on the finite-dimensionality of topological products of compacta for the large inductive dimension.

Let X be a non-empty set. We introduce semi-convergence classes on X in order to obtain a modification of classical Kelley's theorem. Subsequently, we do some further investigations on ideal convergence classes (see [7] ). Finally, we introduce ideal semi-convergence classes C ′ on X, in order to ensure the existence of a unique topology τ on X such that: a net ( s d ) d ∈ D I -semi-convergences ( C ′ ) to x ∈ X i.e. ( ( s d ) d ∈ D , x , I ) ∈ C ′ , where I is an ideal of D, if and only if ( s d ) d ∈ D I -converges to x relative to the topology τ.

We prove that if H is a topological group such that all closed subgroups of H are separable, then the product G × H has the same property for every separable compact group G. Let c be the cardinality of the continuum. Assuming 2 ω 1 = c , we show that there exist: • pseudocompact topological abelian groups G and H such that all closed subgroups of G and H are separable, but the product G × H contains a closed non-separable σ-compact subgroup; • pseudocomplete locally convex vector spaces K and L such that all closed vector subspaces of K and L are separable, but the product K × L contains a closed non-separable σ-compact vector subspace.

In this work we establish that finite directed graphs give rise to semiflows on the power set of their nodes. We analyze the topological dynamics for semiflows on finite directed graphs by characterizing Morse decompositions, recurrence behavior and attractor-repeller pairs under weaker assumptions. As is expected, the discrete metric plays an important role in our constructions and their consequences. The connections between the semiflow, graph theory and Markov chains are here explored. We lay the foundation for a dynamical systems approach to hybrid systems with Markov chain type perturbations., Comment: 29 pages

In this paper we continue the investigations of the relationship between T 0 -quasi-metric spaces and partially ordered metric spaces. Among other things, we establish an equivalence of categories between so-called maximal T 0 -quasi-metric spaces and partially ordered metric spaces produced by T 0 -quasi-metrics. In the linear context we give geometric interpretations of the obtained results. In particular, we also derive a representation theorem for injective asymmetrically normed spaces.

Every group G carries an intrinsically defined (by means of solution sets of one-variable equations) topology Z G , named Zariski topology. It is related to another topology M G having as closed sets all unconditionally closed sets of G, named Markov topology, after Markov who implicitly introduced both topologies in dealing with a series of problems related to group topologies. The aim of this survey is to enlighten the utility of these topologies in resolving Markov problems, as well as other challenging problems in the area of topological groups, mainly related to topologization via group topologies with certain properties. We show that these topologies shelter under the same umbrella as distant issues as abelian groups and highly non-abelian ones, as permutation groups and homeomorphism groups.

In this paper, after discussing the one point compactification of the Khalimsky line (resp. the Khalimsky plane), denoted by ( Z ⁎ , κ ⁎ ) (resp. ( ( Z 2 ) ⁎ , ( κ 2 ) ⁎ ) ), we study various properties of these compactifications associated with the semi- T 1 2 axiom, a non-Alexandroff structure, a non-cut-point space and so forth. We also investigate dense subsets and nowhere dense subsets of ( Z ⁎ , κ ⁎ ) and ( ( Z 2 ) ⁎ , ( κ 2 ) ⁎ ) . Finally, motivated by a particular point topology and an excluded point topology, we develop two kinds of new topologies as quotient topological spaces of ( Z ⁎ , κ ⁎ ) called an excluded two points topology and a cofinite particular point topology.

For a completely regular space X , let C B ( X ) be the normed algebra of all bounded continuous scalar-valued mappings on X equipped with pointwise addition and multiplication and the supremum norm and let C 0 ( X ) be its subalgebra consisting of mappings vanishing at infinity. For a non-vanishing closed ideal H of C B ( X ) we study properties of its spectrum sp ( H ) which may be characterized as the unique locally compact (Hausdorff) space Y such that H and C 0 ( Y ) are isometrically isomorphic. We concentrate on compactness properties of sp ( H ) and find necessary and sufficient (algebraic) conditions on H such that the spectrum sp ( H ) satisfies (topological) properties such as the Lindelof property, σ -compactness, countable compactness, pseudocompactness and paracompactness.

The concept of cut point convex sets is used to study non-cut points of a connected topological space. Some relations between cut point convex sets, H-sets and COTS are established. We prove that an H-set of a COTS is contained in a COTS with endpoints a and b for some a, b in the H-set. It is shown that if a connected topological space X has at most two non-cut points and an R ( i ) set that contains all the closed cut points of X, then X is a COTS with endpoints. Further we show that if every non-degenerate proper regular closed connected subset of a connected topological space X contains only finitely many closed points of X, then X has at least two non-cut points. A characterization of a non-indiscrete finite connected subspace of the Khalimsky line is also obtained.

A space X is selectively absolutely star-Lindelof [1] , [3] if for each open cover U of X and any sequence ( D n : n ∈ ω ) of dense subsets of X, there are finite sets F n ⊆ D n ( n ∈ ω ) such that S t ( ⋃ n ∈ ω F n , U ) = X . In this paper, we continue to investigate topological properties of selectively absolute star-Lindelof spaces, and show the following statements: (1) There exists a Tychonoff selectively a-star-Lindelof, pseudocompact space X having a regular closed G δ subset which is not star-Lindelof (hence not selectively a-star-Lindelof); (2) Assuming 2 ℵ 0 = 2 ℵ 1 , there exists a normal selectively a-star-Lindelof space X having a regular closed G δ subset which is not star-Lindelof (hence not selectively a-star-Lindelof); (3) An open F σ -subset of a selectively a-star-Lindelof space is selectively a-star-Lindelof; (4) For any cardinal κ, there exists a Tychonoff selectively a-star-Lindelof, pseudocompact space X such that e ( X ) ≥ κ .

Let α be an involution of a Peano continuum X with nowhere dense fixed point set. Let α ⁎ be the induced involution on the hyperspace 2 X of nonempty closed subsets of X topologized by a Hausdorff metric. Let E ⊆ 2 X be a non-degenerate, α ⁎ -invariant hyperspace of X that is an inclusion or growth hyperspace in the sense of Curtis and Schori, and assume that the complement of { X } in E is contractible. Let S ( E ) be its fixed point set. If E is an inclusion hyperspace, then the restriction α ˆ ⁎ of α ⁎ to E is conjugate with the involution i d × τ of the Hilbert cube S ( E ) × Π i ≥ 1 I i , where τ is the involution of Π i ≥ 1 I i that reflects each coordinate across its mid-point. If E is a growth hyperspace of X and X contains no open subset homeomorphic to an arc, then the same result holds. In either case, if the complement of { X } in S ( E ) is contractible, then α ˆ ⁎ is conjugate with the involution σ of Π i ≥ 1 I i that reflects each even coordinate across its mid-point.

Hindman and Leader first introduced the notion of Central sets near zero for dense subsemigroups of $((0,\infty),+)$ and proved a powerful combinatorial theorem about such sets. Using the algebraic structure of the Stone-$\breve{C}$ech compactification, Bayatmanesh and Tootkabani generalized and extended this combinatorial theorem to the central theorem near zero. Algebraically one can define quasi-central set near zero for dense subsemigroup of $((0,\infty),+)$, and they also satisfy the conclusion of central sets theorem near zero. In a dense subsemigroup of $((0,\infty),+)$, C-sets near zero are the sets, which satisfies the conclusions of the central sets theorem near zero. Like discrete case, we shall produce dynamical characterizations of these combinatorically rich sets near zero., Comment: 12 pages

Let C ( X ) be the hyperspace of subcontinua of a continuum X. A Whitney block is a set of the form μ − 1 ( [ s , t ] ) , where μ : C ( X ) → [ 0 , 1 ] is a Whitney map and 0 ≤ s t ≤ 1 . In this paper, we study the following implication: if X has property P, then each Whitney block in C ( X ) has property P. We consider the following properties: connectedness im kleinen, being absolute neighborhood retract, local contractibility, and m-mutual aposyndesis.

A space X is called a k R -space, if X is Tychonoff and the necessary and sufficient condition for a real-valued function f on X to be continuous is that the restriction of f to each compact subset is continuous. In this paper, we discuss the k R -property of products of sequential fans and free Abelian topological groups by applying the κ -fan introduced by Banakh. In particular, we prove the following two results: (1) The space S ω 1 × S ω 1 is not a k R -space. (2) The space S ω × S ω 1 is a k R -space if and only if S ω × S ω 1 is a k -space if and only if b > ω 1 . These results generalize some well-known results on sequential fans. Furthermore, we generalize some results of Yamada on the free Abelian topological groups by applying the above results. Finally, we pose some open questions about the k R -spaces.

For a discrete group G, we use the natural correspondence between ideals in the Boolean algebra P G of subsets of G and closed subsets in the Stone–Cech compactification βG as a right topological semigroup to introduce and characterize some new ideals in βG. We show that if a group G is either countable or Abelian then βG contains no ideals that are maximal among the closed proper subideals of G ⁎ , G ⁎ = β G ∖ G , but this statement does not hold for the group S κ of all permutations of an infinite cardinal κ. We characterize the minimal closed ideal in βG containing all idempotents of G ⁎ .

L. Zajicek in 1986 and V. Kelar in 1990 defined topologies p and s generated by the notions of porosity and strong porosity. Applying these notions, we described maximal additive classes for porouscontinuous functions S 0 , M 1 and P 0 defined by J. Borsik and J. Holos in 2014. Furthermore, we defined new family of topologies generated by porosity and strong porosity, which are used to studying maximal multiplicative classes for porouscontinuous functions S 0 and M 1 . Some other properties of superporosity and strong superporosity are considered.

Inverse limit spaces of compacta with upper semi-continuous compact set-valued functions are studied via shift maps and their variants. We give a representation of such spaces as the limits of ordinary inverse sequences, which allows us to prove some known results and their extensions in a unified scheme. Next we present a scheme to construct shift dynamics on the inverse limit space with various dynamical features. In particular we construct an inverse sequence over [ 0 , 1 ] with a single upper semi-continuous function f as its bonding function such that (i) the inverse limit space [ 0 , 1 ] f is homeomorphic to the Cantor set and (ii) the shift map σ f : [ 0 , 1 ] f → [ 0 , 1 ] f is topologically conjugate to a minimal subshift of a Bernoulli full shift. Also we study local/global connectivity of the inverse limit space over a compactum with a single upper semi-continuous bonding function in terms of homotopy/(co)homology groups, again via shift maps and their variants.

Given a metric space X = ( X , d ) we show in ZF that: (a) The following are equivalent: (i) For every two closed and disjoint subsets A , B of X, d ( A , B ) > 0 . (ii) Every countable open cover of X has a Lebesgue number. (iii) Every real valued continuous function on X is uniformly continuous. (iv) For every countable (resp. finite, binary) open cover U of X, there exists a δ > 0 such that for all x , y ∈ X with d ( x , y ) δ , { x , y } ⊆ U for some U ∈ U . (b) If X is connected then: X is countably compact iff every open cover of X has a Lebesgue number iff for every two closed and disjoint subsets A , B of X, d ( A , B ) > 0 .

We introduce large scale analogues of topological monotone and light maps, which we call coarsely monotone and coarsely light maps respectively. We show that these two classes of maps constitute a factorization system on the coarse category. We also show how coarsely monotone maps arise from a reflection in a similar way to classically monotone maps, and prove that coarsely monotone maps are stable under those pullbacks which exist in the coarse category. For the case of maps between proper metric spaces, we exhibit some connections between the coarse and classical notions of monotone and light using the Higson corona. Finally, we look at some coarse properties which are preserved by coarsely light maps such as finite asymptotic dimension and exactness, and make some remarks on the situation for groups and group homomorphisms.

In this paper, sufficient conditions for the direct sum of countable linear operators on Banach spaces to be Li–Yorke chaotic (distributionally chaotic) are presented. These conditions enable us to construct a densely distributionally chaotic direct sum operator such that none of its factor operators exhibits Li–Yorke chaos. As an application, it is shown that for any b > a > 0 , there exists an invertible operator T acting on a Hilbert space such that [ a , b ] = { λ > 0 : λ T is distributionally chaotic } and for any distinct λ 1 , λ 2 ∈ [ a , b ] , the operators λ 1 T and λ 2 T have no common irregular vectors.

The authors [5] defined cell structures to be inverse sequences of graphs with some mild convergence conditions and they defined cell maps between cell structures. They showed how to obtain from these all complete metric spaces and continuous mappings between such spaces. In this paper that work is extended to the class of topologically complete spaces. This shows that topologically complete spaces and their continuous functions are determined by discrete approximations.

Let M n be a closed connected finite dimensional manifold. The suspension Σ M n of M n is homogeneous if and only if M n is homeomorphic to a topological sphere S n . Furthermore, Σ M n is 1 2 -homogeneous if and only if M n is not homeomorphic to S n .

We show that the Cartesian product of the arc and a solenoid has the fupcon property, therefore answering a question raised by Illanes. This combined with Illanes' result implies that the product of a Knaster continuum and a solenoid has the fupcon property, therefore answering a question raised by Bellamy and Łysko in the affirmative. Finally, we show that a product of two Smith's nonmetric pseudo-arcs has the fupcon property.

In 2012, Žiga Virk introduced the notion of an extensional equivalence, herein called an extensional map, and used it to generalize part of the extension theory factorization theorem of M. Levin, L. Rubin, and P. Schapiro. Here were are going to study this notion in the setting of inverse systems of compact Hausdorff spaces and approximate inverse systems of compact metric spaces. In both cases we will show that given a surjective map f to the limit, if each coordinate map p γ ∘ f is an extensional map, then so is f.

Balogh proved that PFA , the Proper Forcing Axiom, implies that a countably compact space with countable tightness is either compact or contains an uncountable free sequence. Eisworth established the relevance to proper forcing of strengthening the countable tightness assumption to that of hereditary countable π-character. Eisworth proved that for any countably compact space with hereditary countable π-character there is a totally proper forcing that adds an uncountable free sequence. We extend these results by showing that PFA implies that countably compact spaces are closed in spaces that have hereditary countable π-character. This gives a countably compact version of the Moore–Mrowka problem in the class of spaces with hereditary countable π-character.

In this article the new properties of relative retracts in the context of relative homotopy are studied. The results of the studies are particularly applied to the characterization of connected and locally connected spaces and absolute neighborhood retracts.

A Heegaard splitting V 1 ∪ S V 2 is called weakly keen with distance 2 if there are essential separating disks D i ⊂ V i such that a component of V i − D i is homeomorphic to F i × I and there is an unique geodesic { ∂ D 1 , C 0 , ∂ D 2 } in C ( S ) connecting ∂ D 1 to ∂ D 2 , where C 0 is an essential simple closed curve in S and F i is a component of ∂ − V i for i = 1 , 2 . In this paper, we give a sufficient condition for the weakly keen Heegaard splittings to be keen. At last, we give a sufficient condition for the self-amalgamation of a weakly keen Heegaard splitting to be unstabilized.

A Lagrangian star is a system of three Lagrangian submanifolds of the symplectic space intersecting at a common point. In this work we classify transversal Lagrangian stars in the symplectic space in the analytic category under the action of symplectomorphisms by using the method of algebraic restrictions. We present a list of all transversal Lagrangian star.

The multiplication of a semitopological (quasitopological) group G is called sequentially continuous if the product map of G × G into G is sequentially continuous. In this paper, we mainly consider the properties of semitopological (quasitopological) groups with sequentially continuous multiplications and three-space problems in quasitopological groups. It is showed that (1) every snf -countable semitopological group G with the sequentially continuous multiplication is sof -countable; (2) if G is a sequential quasitopological group with the sequentially continuous multiplication, then G contains a closed copy of S ω if and only if it contains a closed copy of S 2 , which give a partial answer to a problem posed by R.-X. Shen; (3) let G be a quasitopological group with the sequentially continuous multiplication, then the following are equivalent: (i) G is a sequential α 4 -space; (ii) G is Frechet; (iii) G is strongly Frechet; (4) (MA+¬CH) there exists a non-metrizable, separable, normal and Moore quasitopological group; (5) some examples are constructed to show that metrizability, first-countability and second-countability are not three-space properties in the class of quasitopological groups.

In this paper, we continue the study of the symmetric products of generalized metric spaces in [39] . We consider the topological properties P such that the n-fold symmetric product F n ( X ) of a topological space X has the topological properties P if and only if the space X or the product X n does for each or some n ∈ N . Depending on the operations under closed subspaces, finite products and closed finite-to-one mappings, two general stability theorems are obtained on symmetric products. We can apply the methods to unify and simplify the proofs of some old results in the literature and obtain some new results on symmetric products, list or prove 68 topological properties which satisfy the general stability theorems, and give answers to Questions 3.6 and 3.35 in [39] .

We describe how to attach a weighted graph to each stable map from closed surfaces to projective plane and prove that any weighted graph with non negatively weighted vertices is the graph of some stable map from a closed surface to the projective plane.

In this article, we propose some conditions on the modified defect relations of the Gauss map of a complete minimal surface $M$ to show that $M$ has finite total curvature., 22 pages. the references and the introduction were changed. arXiv admin note: substantial text overlap with arXiv:1710.02023; text overlap with arXiv:1411.2730

In this paper, we prove that DCCC is self-dual with respect to neighborhood assignments. Some related results on dually CCC (or dually DCCC) spaces are also obtained. Moreover, we prove that the cardinality of a dually CCC space X is at most 2 ω if X satisfies one of the following conditions: (1) X has a rank 2-diagonal; (2) X is first countable and has a G δ -diagonal; (3) X is first countable and perfect. Finally, we prove that the extent of a first countable dually CCC space is at most 2 ω .

It is a simple fact that a subgroup generated by a subset $A$ of an abelian group is the direct sum of the cyclic groups $\langle a\rangle$, $a\in A$ if and only if the set $A$ is independent. In [5] the concept of an $independent$ set in an abelian group was generalized to a $topologically$ $independent$ $set$ in a topological abelian group (these two notions coincide in discrete abelian groups). It was proved that a topological subgroup generated by a subset $A$ of an abelian topological group is the Tychonoff direct sum of the cyclic topological groups $\langle a\rangle$, $a\in A$ if and only if the set $A$ is topologically independent and absolutely Cauchy summable. Further, it was shown, that the assumption of absolute Cauchy summability of $A$ can not be removed in general in this result. In our paper we show that it can be removed in precompact groups. In other words, we prove that if $A$ is a subset of a {\em precompact} abelian group, then the topological subgroup generated by $A$ is the Tychonoff direct sum of the topological cyclic subgroups $\langle a\rangle$, $a\in A$ if and only if $A$ is topologically independent. We show that precompactness can not be replaced by local compactness in this result.

An abelian group N with discrete topology is called cancellable if for any two abelian topological groups G and H, the product group G × N is topologically isomorphic to H × N if and only if G and H are topologically isomorphic. In this paper we show that the additive group Z of integers is cancellable which answers a problem posed in [1] negatively. We also show that every finitely generated abelian group is cancellable. Moreover, we show that a divisible group D is cancellable if and only if the maximal torsion-free subgroup of D is the direct sum of a finite number of copies of the rationals and for each prime p, the p-primary component of D is the direct sum of a finite number of copies of the quasi-cyclic group Z ( p ∞ ) .

A Hausdorff compact space is called a Fedorchuk compactum (or F-compactum) if it admits a decomposition into a special well-ordered inverse system with fully closed neighboring projections. It is known that the product of fully closed mappings is not fully closed, as a rule. We prove the same property for the class of Fedorchuk compacta: the product of F-compacta of spectral height 3 is never an F-compactum of countable spectral height.

For a metric continuum X and a positive integer n , we consider the hyperspaces C n ( X ) (respectively, F n ( X ) ) of all nonempty closed subsets of X having at most n components (respectively, n points). Given positive integers n and m such that n ≥ m , we define H S m n ( X ) as the quotient space C n ( X ) / F m ( X ) which is obtained from C n ( X ) by shrinking F m ( X ) to a point. In this paper we prove that if X is a finite graph and Y is a continuum such that H S m n ( X ) is homeomorphic to H S m n ( Y ) , then X is homeomorphic to Y .

A classical theorem of Adams, Harris, and Switzer states that the 0th grading of complex $K$-theory cooperations, $KU_0ku$ is isomorphic to the space of numerical polynomials. The space of numerical polynomials has a basis provided by the binomial coefficient polynomials, which gives a basis of $KU_0ku$. In this paper, we produce a new $p$-local basis for $KU_0ku_{(p)}$ using the Adams splitting. This basis is established by using well known formulas for the Hazewinkel generators. For $p=2$, we show that this new basis coincides with the classical basis modulo higher Adams filtration., Comment: 1 figure

We provide a general formula and give an explicit expression of the Lefschetz zeta function for any quasi-unipotent map on the space X = S l × ⋯ × ︸ n − times S l , with l > 1 . Among the quasi-unipotent maps are Morse–Smale diffeomorphisms. The Lefschetz zeta function is used to characterize the minimal set of Lefschetz periods for Morse–Smale diffeomorphisms on X ; we completely describe this set, for families containing infinitely many Morse–Smale diffeomorphisms. The results of the present article are based on the techniques used in [5] , in the computation of the Lefschetz zeta function for quasi-unipotent self maps on the n -dimensional torus.

It is shown that the Bolzano property characterizes the covering dimension. Some relationships between the Steinhaus chain property and a dimension are discussed. Finally a new dimension for arbitrary topological spaces is defined.

Given compact spaces X and Y, if X is Eberlein compact and C p , n ( X ) is homeomorphic to C p , n ( Y ) for some natural n, then Y is also Eberlein compact; this result answers a question posed by Tkachuk. Assuming existence of a Souslin line, we give an example of a Corson compact space with a Lindelof subspace that fails to be Lindelof Σ; this gives a consistent answer to another question of Tkachuk. We establish that every Σ s -product of K-analytic spaces is Lindelof Σ and C p ( X ) is a Lindelof Σ-space for every Lindelof Σ-space X contained in a Σ s -product of real lines. We show that C p ( X ) is Lindelof for each Lindelof Σ-space X contained in a Σ-product of real lines. We prove that C p ( X ) has the Collins–Roscoe property for every dyadic compact space X and generalize a result of Tkachenko by showing, with a different method, that the inequality w ( X ) ≤ n w ( X ) N a g ( X ) holds for regular spaces.

Given a metric continuum X, we define the hyperspace of regular subcontinua of X as the collection of all regular closed subcontinua of X. This is a new hyperspace for X. In this paper we study the connectedness, compactness, arcwise connectedness and contractibility of this hyperspace. Also we raise some open questions.

A space is said to be 1 n -homogeneous provided there are exactly n orbits for the action of the group of homeomorphisms of the space onto itself. In this paper, we investigate 1 n -homogeneity in suspensions and cones of locally compact, homogeneous and finite dimensional metric spaces, we prove that if X is a solenoid, then the hyperspace of all subcontinua of X, is 1 3 -homogeneous. Moreover, we determine conditions under which the 2-nd cone of a Hausdorff space is 1 2 -homogeneous. Finally, we include a list of open problems related to this topic.

Properties weaker than monotone countable metacompactness are studied in PIGO and compact spaces. GO-spaces with a σ -closed-discrete dense subset and the N Z ( ω ) -property are metrizable generalizing results of Bennett, Hart and Lutzer and Peng and Li on monotonically countably metacompact spaces. NSR pair-families are introduced, and pair-bases that are NSR pair-families or countable unions of such pair-families are studied. By modifying results of Chase and Gruenhage, we show if a space X with a σ -NSR pair-base is compact, then X is metrizable, generalizing recent results by Chase and Gruenhage and some older results of Gruenhage and Nyikos. We show PIGO spaces with a σ -closed-discrete dense subset and a σ -NSR pair-base are metrizable. Relationships between these properties and others in the literature are also established.