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In this paper we work toward the HOMFLYPT skein module of $L(p, 1)$, $\mathcal{S}(L(p,1))$, via braids. Our starting point is the linear Turaev-basis, $\Lambda^{\prime}$, of the HOMFLYPT skein module of the solid torus ST, $\mathcal{S}({\rm ST})$, which can be decomposed as the tensor product of the "positive" ${\Lambda^{\prime}}^+$ and the "negative" ${\Lambda^{\prime}}^-$ sub-modules, and the Lambropoulou invariant, $X$, for knots and links in ST, that captures $S({\rm ST})$. It is a well-known result by now that $\mathcal{S}(L(p, 1))=\frac{\mathcal{S}(ST)}{}$, where bbm's (braid band moves) denotes the isotopy moves that correspond to the surgery description of $L(p, 1)$. Namely, a HOMFLYPT-type invariant for knots and links in ST can be extended to an invariant for knots and links in $L(p, 1)$ by imposing relations coming from the performance of bbm's and solving the infinite system of equations obtained that way. \smallbreak In this paper we work with a new basis of $\mathcal{S}({\rm ST})$, $\Lambda$, and we relate the infinite system of equations obtained by performing bbm's on elements in $\Lambda^+$ to the infinite system of equations obtained by performing bbm's on elements in $\Lambda^-$ via a map $I$. More precisely we prove that the solutions of one system can be derived from the solutions of the other. Our aim is to reduce the complexity of the infinite system one needs to solve in order to compute $\mathcal{S}(L(p,1))$ using the braid technique. Finally, we present a generating set and a potential basis for $\frac{\Lambda^+}{}$ and thus, we obtain a generating set and a potential basis for $\frac{\Lambda^-}{}$. We also discuss further steps needed in order to compute $\mathcal{S}(L(p,1))$ via braids., Comment: 26 pages, 6 figures. arXiv admin note: text overlap with arXiv:1802.09376, arXiv:1702.06290

Let K be a nontrivial knot in S 3 with the exterior E ( K ) , and denote π 1 ( E ( K ) ) by G ( K ) . We prove that for any hyperbolic knot K and any nontrivial element g ∈ G ( K ) , there are only finitely many Dehn fillings of E ( K ) which trivialize g. We also demonstrate that there are infinitely many nontrivial elements in G ( K ) which cannot be trivialized by nontrivial Dehn fillings.

A coherent band surgery is a local move of an oriented knot or link by doing a surgery along an added band. The coherent band-Gordian distance between two links is the minimum number of coherent band surgeries needed to transform one into the other. We give a table of coherent band-Gordian distances between a knot and 2-component link with up to seven crossings.

An embedded/immersed surface-knot is a closed and connected surface embedded/immersed in R 4 , respectively. The triple point number t ( F ) of an embedded/immersed surface-knot F is the minimum number of triple points required for a diagram of F. Satoh proved that (i) there does not exist an embedded surface-knot F such that t ( F ) = 1 , and (ii) there does not exist an embedded 2-knot F such that t ( F ) = 2 or 3 . In this paper, we prove similar results for immersed surface-knots with some conditions.

A cubical 2-knot K 2 is an embedding of the 2-sphere in the 2-skeleton of the canonial cubulation of R 4 , so K 2 is the union of m ( K 2 ) unit squares, hence its area is m ( K 2 ) . In [6] was proved that given two cubical knots of dimension two in R 4 , they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. We say that a cubical 2-knot K 2 is weakly minimal if the area of K 2 can not be reduced by any single cubical move. One interesting question is the following: Given a knot type, what is the area needed for a cubical 2-knot on the canonical cubulation of R 4 to be a weakly minimal knot with the given knot type? In this paper, we answer this question for the spun trefoil knot.

Let E ( X ) be the set of based homotopy classes of based self-homotopy equivalences of a CW-complex X. The concept of E ( X ) is applied to the category of pairs and is extended to a general concept E ( α ) for a map α : A → B . In this study, E ( α ) is generalized to E γ ( α ) for two objects α and γ. Several generalized subgroups of E ( α ) or E γ ( α ) are obtained and are combined to form an exact sequence. The exactness and the split property of this sequence is investigated. In particular, the sequence of a product space or a wedge space is demonstrated to be a split exact sequence. The split property and the exactness are used to completely compute those subgroups.

We study the mod p homology of the gauge group G k ( F 4 ) associated with a principal F 4 bundle by computing the Serre spectral sequence for the fibration Ω 0 4 ( F 4 ) → G k ( F 4 ) → F 4 . The main result is that the Serre spectral sequence collapses at the E 2 -term except for p = 2 , 13 .

We establish a general theorem for connectedness of the inverse limit X of an inverse sequence { X i , f i } i ≥ 1 on metric continua with surjective upper semi-continuous set-valued bonding functions, where for each i ≥ 1 , f i has a connected graph, and either f i or f i − 1 is a union of continuum-valued functions. Properties of certain set-valued functions from the factor spaces onto partial graphs in the inverse sequence imply connectedness of X.

This paper explains a research on the character varieties and the Kauffman bracket skein module (KBSM) of knot exteriors, which has been done by the author in his stay at the University of California, Riverside, 2004-2006. As a main consequence, for any 2-bridge knot, we gave a representation theoretical proof to the conjecture that degree 0 abelian knot contact homology H C 0 a b ( K ) of a knot K in the 3-space R 3 is isomorphic to the S L 2 ( C ) -character ring of the fundamental group of the 2-fold branched cover of the 3-sphere S 3 branched along K. (This result was first shown by L. Ng [23] by using cord formalism.) Based on this result, deep studies of the conjecture has been done in [18] , [19] , [20] .

Abe and Tange constructed a sequence of slice disks with the same exterior. Moreover, they showed that these slice disks are ribbon disks. We call the boundaries of the ribbon disks Abe-Tange's ribbon knots. In this paper, we classify Abe-Tange's ribbon knots completely by the Γ-polynomial. Therefore, we see that all ribbon disks in the sequence are mutually inequivalent.

We prove that twisting any quasi-alternating link $L$ with no gaps in its Jones polynomial $V_L(t)$ at the crossing where it is quasi-alternating produces a link $L^{*}$ with no gaps in its Jones polynomial $V_{L^*}(t)$. This leads us to conjecture that the Jones polynomial of any prime quasi-alternating link, other than $(2,n)$-torus links, has no gaps. This would give a new property of quasi-alternating links and a simple obstruction criterion for a link to be quasi-alternating. We prove that the conjecture holds for quasi-alternating Montesinos links as well as quasi-alternating links with braid index 3., 11 pages

Let G be a locally compact first countable Abelian topological group of cardinality ≤ c and suppose that for every n ∈ N the subgroup nG is not totally bounded. We show that (1) the topology of G can be extended to a locally compact first countable semigroup topology T on S = G ⊕ ( ⊕ λ Z + ) for some λ ∈ [ p , c ] such that G is dense in T and ( S , T ) is absolutely closed in the class of cancellative topological semigroups with the Frechet-Urysohn property, and (2) assuming Martin's Axiom, the topology of G can be extended to a locally compact first countable semigroup topology T on S = G ⊕ ( ⊕ c Z + ) such that G is dense in T and ( S , T ) is absolutely closed in the class of all topological semigroups with the Frechet-Urysohn property.

A continuum is a compact connected metric space. A non-empty closed subset B of a continuum X does not block x ∈ X ∖ B provided that the union of all subcontinua of X containing x and contained in X ∖ B is a dense subset of X. The collection of all non-empty closed subsets B of X such that B does not block each element of X ∖ B is denoted by NB ( F 1 ( X ) ) . In this paper, we find conditions under which NB ( F 1 ( X ) ) and the hyperspace of non-weak cut sets NWC ( X ) coincide and we exhibit a dendroid X for which NWC ( X ) is a non-empty proper subset of NB ( F 1 ( X ) ) . Also, we present geometric models for NB ( F 1 ( X ) ) ; particularly, some of them give examples for a question posed by Escobedo, Lopez and Villanueva in 2012. Finally, we prove that there exists a family of continua X such that the collection of hyperspaces NB ( F 1 ( X ) ) is an uncountable incomparable family of continua.

We introduce a new functor R ˜ from p r o ⁎ - G r p to p r o - G r p and use it to give a new characterisation of coarse shape groups. It is shown that lim ← ⁡ ∘ R ˜ ∘ p r o ⁎ - π k = π ˇ k ⁎ , thus R ˜ ∘ p r o ⁎ - π k is a full analogue of p r o - π k . Furthermore, the Hurewicz theorem in a new context is proposed. We complete with a corollary stating isomorphicity of the first nontrivial coarse shape group and coarse shape homology group of a pointed continuum, assertion that does not hold for shape groups.

The class of weakly linearly Lindelof spaces was introduced and studied by Juhasz, Tkachuk and Wilson in [7] . Recall that a space X is weakly linearly Lindelof if for any family U of non-empty open subsets of X of regular uncountable cardinality κ, there exists a point x ∈ X such that every neighborhood of x meets κ-many elements of U . In this paper, we show that: (1) If X is a weakly linearly Lindelof space and U is an open cover of X, then for the open cover { St 2 ( x , U ) : x ∈ X } of X, there exists a countable subset A ⊂ X such that St 2 ( A , U ) ‾ = X ; (2) Every weakly linearly Lindelof normal metaLindelof space is weakly Lindelof; (3) If X is a first countable regular space, then M ( X ) (generated by Moore Machine) is weakly linearly Lindelof if and only if X is weakly linearly Lindelof; (4) Every product of a weakly linearly Lindelof space and a space of countable spread (or a separable space) is weakly linearly Lindelof; (5) If a subspace X ⊂ ω 1 ω is weakly linearly Lindelof, then X is second countable (and hence, metrizable); (6) If X is a weakly linearly Lindelof Baire space with a rank 2-diagonal such that w e ( X ) ≤ ω 1 , then | X | ≤ c ; (7) The space X is cellular-WLL if and only if it is weakly linearly Lindelof.

We give a family of metrizable topological groups which are not strictly ω-balanced. This answers a question posed in the following paper: Xie and Jin (2019) [12] .

The notion of d-ideal has been abstracted by Martinez and Zenk from Riesz spaces to algebraic frames. They introduced the d-nucleus and d-elements. In this paper we extend several characterizations of d-elements that parallel similar characterizations of d-ideals in rings. For instance, calling a coherent map between algebraic frames “weakly skeletal” if it maps compact elements with equal pseudocomplements to images with equal pseudocomplements, we show that an a ∈ L is a d-element if and only if a = h ⁎ ( 0 ) for some weakly skeletal map h : L → M , where h ⁎ denotes the right adjoint of h. The sublocale of L induced by the d-nucleus is denoted by dL. We characterize when d ( L ⊕ M ) ≅ d L ⊕ d M . We weaken the notion of d-element by defining eL to be the set of elements that are joins of double pseudocomplements of compact elements. We show that if L = d L and M = d M , then L ⊕ M = e ( L ⊕ M ) . Clearly, d L ⊆ e L . We give an example to show that the containment can be proper. Finally, we show that dL is always a sublocale of eL.

Let X be a convex set in a vector space, Y be a nonempty set and S , T : X ⇉ Y two set-valued mappings. S is said to be a weak KKM mapping w.r.t. T if for each nonempty finite subset A of X and any x ∈ conv A , T ( x ) ∩ S ( A ) ≠ ∅ . Recently, the authors obtained two intersection theorems for a pair of such mappings, when X is a compact convex subset of a topological vector space. In this paper, we obtain open versions of the above mentioned theorems when X is a compact convex set in R n . As applications, we establish several minimax inequalities and existence criteria for the solutions of three types of set-valued equilibrium problems.

Bal and Kocinac in [2] introduced and studied the class of selectively star-ccc spaces. A space X is called selectively star-ccc if for every open cover U of X and for every sequence ( A n : n ∈ ω ) of maximal pairwise disjoint open families in X, there exists a sequence ( A n : n ∈ ω ) such that A n ∈ A n for every n ∈ ω and St ( ⋃ n ∈ ω A n , U ) = X . In this paper, we show that there exists a Tychonoff selectively 2-star-ccc space which is neither strongly star Lindelof nor selectively star-ccc, which gives a positive answer to a question of Bal and Kocinac [2] . Under 2 ℵ 0 = 2 ℵ 1 , we even provide a normal example of a selectively 2-star-ccc space which is neither strongly star Lindelof nor selectively star-ccc. Finally, we prove that every open F σ -subset of a selectively star-ccc space is selectively star-ccc. Some new questions are also posed.

Let ( S , + ) be an infinite commutative semigroup with identity 0. Let u , v ∈ N and let A be a u × v matrix with nonnegative integer entries. If S is cancellative, let the entries of A come from Z . Then A is image partition regular over S ( I P R / S ) iff whenever S ∖ { 0 } is finitely colored, there exists x → ∈ ( S ∖ { 0 } ) v such that the entries of A x → are monochromatic. The matrix A is centrally image partition regular over S ( C I P R / S ) iff whenever C is a central subset of S, there exists x → ∈ ( S ∖ { 0 } ) v such that A x → ∈ C u . These notions have been extensively studied for subsemigroups of ( R , + ) or ( R , ⋅ ) . We obtain some necessary and some sufficient conditions for A to be I P R / S or C I P R / S . For example, if G is an infinite divisible group, then A is C I P R / G iff A is I P R / Z . If for all c ∈ N , c S ≠ { 0 } and A is I P R / N , then A is I P R / S . If S is cancellative, c ∈ N , and c S = { 0 } , we obtain a simple sufficient condition for A to be I P R / S . It is well-known that A is I P R / S if A is a first entries matrix with the property that cS is a central⁎ subset of S for every first entry c of A. We extend this theorem to first entries matrices whose first entries may not satisfy this condition. We discuss whether, if S is finitely colored, there exists x → ∈ ( S ∖ { 0 } ) v , with distinct entries, for which the entries of A x → are monochromatic and distinct. Along the way, we obtain several new results about the algebra of βS, the Stone-Cech compactification of the discrete semigroup S.

We provide an example to show that the monotone Sokolov property is not necessarily preserved under compact continuous images. Furthermore, we prove that if X 2 ∖ Δ X is either monotonically Sokolov or monotonically retractable, then X must be cosmic; and that if X is either hereditarily monotonically Sokolov or hereditarily monotonically retractable, then X has a countable network. Moreover, we characterize the Lindelof Σ-property in C p -spaces on Alexandrov doubles, and show that the L Σ ( L Σ ( ⩽ ω ) ) -property is equivalent to the L Σ ( ⩽ ω ) -property for the class of C p -spaces and for the class of Gul'ko spaces. These results solve some problems published by Kalenda [7] , Tkachuk [23] , Garcia-Ferreira and Rojas-Hernandez [5] , and Molina-Lara and Okunev [11] .

We study a class of ideals introduced by Pospisil. We answer a question of the second author by proving that there is an F σ δ σ ideal I such that every filter of character less than c can be extended to an I -ultrafilter. We also prove that this statement is consistently false for F σ δ -ideals.

We continue to study the properties of the class CSS of topological groups in which all closed subgroups are separable. One of the main problems here is whether the class CSS is finitely productive or not. We solve the problem in the negative and present three examples with different combinations of additional properties: (1) there exist in ZFC precompact Abelian groups H and K in CSS such that H × K is not in CSS; (2) under the assumption of 2 ω 1 = 2 ω , there exists a pseudocompact Abelian group G ∈ C S S such that G × G ∉ C S S ; (3) under the assumption of MA & ¬ CH, there exist countably compact Abelian groups G and H in CSS such that G × H ∉ C S S . Our example in (2) improves upon an example presented in the recent article [11] by A. Leiderman and the third listed author, while the example in (3) answers a question raised in that article.

We prove the following consistency result for cardinal sequences of length ω 3 : if GCH holds and λ ≥ ω 2 is a regular cardinal, then in some cardinal-preserving generic extension 2 ω = λ and for every ordinal η ω 3 and every sequence f = 〈 κ α : α η 〉 of infinite cardinals with κ α ≤ λ for α η and κ α = ω if cf ( α ) = ω 2 , we have that f is the cardinal sequence of some LCS space. Also, we prove that for every specific uncountable cardinal λ it is relatively consistent with ZFC that for every α , β ω 3 with cf ( α ) ω 2 there is an LCS space Z such that CS ( Z ) = 〈 ω 〉 α ⌢ 〈 λ 〉 β .

In this paper, results of Tsarelunga resp. Comfort, Szambien and the first-listed author are improved. Throughout this abstract, ( R , T ) denotes a nonmetrizable compact ring. First a main tool is shown: If ( R , T ) is topologically nilpotent, then w ( R ) = w ( R / R 2 ‾ ) holds. By using tensor products of unitary modules it is proved that every nonmetrizable compact ring with an identity has a proper pseudocompact refinement. ( R , T ) admits exactly 2 2 | R | -many pseudocompact ring topologies on R finer than T in the following cases: R is a commutative local ring; ( R , T ) is topologically nilpotent; ( R , T ) is commutative such that w ( R , T ) is a regular cardinal number.

Given a compact abelian group G, we study the poset G δ ( G ) of G δ -dense subgroups of G and the subgroup δ - den ( G ) : = ⋂ G δ ( G ) , named G δ -density nucleus of G. It turns out that for every compact abelian group G: (i) the subgroup δ - den ( G ) is compact metrizable and coincides with the intersection of just two members D 0 , D 1 ∈ G δ ( G ) ; (ii) if δ - den ( G ) = { 0 } , then there exists some independent family F in G δ ( G ) (i.e., with 〈 ⋃ F 〉 = ⨁ i ∈ I D i ) such that: (ii1) | F | = r ( G ) , the free rank of G, if mG is not metrizable for any positive m ∈ N ; otherwise, (ii2) when G is (necessarily) bounded torsion, | F | coincides with the least leading Ulm-Kaplansky invariant of G. (iii) in option (ii1) the members of F can be chosen to be free subgroups of G precisely when G admits a free dense subgroup (equivalently, when r ( G ) ≥ d ( G ) ); F can have (the maximum possible) size | G | if and only if r ( G ) = | G | ; (iv) in option (ii2) the family F can have size | G | if and only if all leading Ulm-Kaplansky invariants of G coincide with | G | . Resuming, the compact abelian groups G with trivial G δ -density nucleus δ - den ( G ) are, roughly speaking, “maximally fragmented” into G δ -dense subgroups. More precisely, the existence of a pair of subgroups D 0 , D 1 ∈ G δ ( G ) with D 0 ∩ D 1 = { 0 } yields the existence of a large independent family F = { D i ∈ I } ⊆ G δ ( G ) with I is as big as possible (i.e., as big as the algebraic structure of G permits in terms of some immediate necessary restraints involving relevant algebraic cardinal invariants of the group, in the spirit of those in (ii)–(iv)).

We study gaps in the lattice of topological group topologies on a given abelian group and compare the properties of the topologies that form a gap. The emphasis is placed on the study of predecessors of the topology of a noncompact LCA group. It is shown that every Hausdorff predecessor, σ, of the topology τ of an LCA group G is finer than the Bohr topology of the group G and that the two topologies, τ and σ, have the same closed subgroups. This implies that the Bohr topology, τ + , of a noncompact LCA group ( G , τ ) is not a predecessor of τ. Complementing these results we prove that all predecessors of the topology of a Hausdorff topological abelian group G are Hausdorff provided that G is torsion free. We also show that if a topological abelian group ( G , τ ) contains a complete subgroup N such that the quotient group G / N is compact, then the predecessors of τ are in a one-to-one correspondence with the predecessors of τ ↾ N on N. In particular, if N is discrete and G / N is compact, then the predecessors of τ are in a one-to-one correspondence with the maximal topological group topologies on N.

We show that if κ ≤ ω and there exists a group topology without non-trivial convergent sequences on an Abelian group H such that H n is countably compact for each n κ then there exists a topological group G such that G n is countably compact for each n κ and G κ is not countably compact. If in addition H is torsion, then the result above holds for κ = ω 1 . Combining with other results in the literature, we show that: a) Assuming c incomparable selective ultrafilters, for each n ∈ ω , there exists a group topology on the free Abelian group G such that G n is countably compact and G n + 1 is not countably compact. (It was already know for ω). b) If κ ∈ ω ∪ { ω } ∪ { ω 1 } , there exists in ZFC a topological group G such that G γ is countably compact for each cardinal γ κ and G κ is not countably compact.

We provide some properties and characterizations of homologically $UV^n$-maps and $lc^n_G$-spaces. We show that there is a parallel between recently introduced by Cauty algebraic $ANR$'s and homologically $lc^n_G$-metric spaces, and this parallel is similar to the parallel between ordinary $ANR$'s and $LC^n$-metric spaces. We also show that there is a similarity between the properties of $LC^n$-spaces and $lc^n_G$-spaces. Some open questions are raised., Comment: 20 pages

Suppose $Y$ is a continuum, $x\in Y$, and $X$ is the union of all nowhere dense subcontinua of $Y$ containing $x$. Suppose further that there exists $y\in Y$ such that every connected subset of $X$ limiting to $y$ is dense in $X$. And, suppose $X$ is dense in $Y$. We prove $X$ is homeomorphic to a composant of an indecomposable continuum, even though $Y$ may be decomposable. An example establishing the latter was given by Christopher Mouron and Norberto Ordo\~nez in 2016. If $Y$ is chainable or, more generally, an inverse limit of identical topological graphs, then we show $Y$ is indecomposable and $X$ is a composant of $Y$. For homogeneous continua we explore similar problems which are related to a 2007 question of Janusz Prajs and Keith Whittington., Comment: 12 pages, 3 figures

We study the compactification of semitopological groups with remainders being locally p ⁎ ⁎ -spaces, and establish: (1) if a semitopological group X has a remainder that is locally a p ⁎ ⁎ -space, then either X is a paracompact p-topological group, or X is meager; (2) if a non-locally compact paratopological group X has a remainder that is locally a p-space, then either X is a Lindelof p-space, or X is σ-compact; (3) if a compact space Z is the union of two non-locally compact semitopological groups X and Y such that Z ∖ X is locally developable, then Z is separable and metrizable; (4) if a non-locally compact Baire semitopological group G has a remainder Y that is the union of a finite family of metrizable subspaces, then G is metrizable. Some results of A.V. Arhangel'skii et al. are improved.

Let S be an orientable closed surface with genus at least two. From S × I , for a given orientation-reversing homeomorphism f from S × { 1 } to S × { 0 } , there is an orientable closed 3-manifold M f = S × I / f which is called a mapping torus. It is known that M f admits a canonical Heegaard splitting H 1 ∪ Σ H 2 . By the construction of Namazi [H.Namazi, Topology Appl. 154 (2007), no. 16, 2939-2949], the mapping class group of this Heegaard splitting, denoted by M o d ( Σ ; H 1 , H 2 ) , contains a reducible mapping class which has infinitely order. So it is interesting to know that for a given element in M o d ( Σ ; H 1 , H 2 ) , whether it is reducible or not. Using the translation length of f in the curve complex, we prove that if f is the identity map or its translation length is at least 8, then each element of M o d ( Σ ; H 1 , H 2 ) is reducible.

Let M be a family of Hausdorff topological groups. A Hausdorff topological group P is called M -cancellable if for any two topological groups G and H in M , the product group G × P is topologically isomorphic to H × P if and only if G and H are topologically isomorphic. If M is the class of all Hausdorff topological groups, M -cancellable topological groups are called cancellable topological groups. We show in this paper that every topological group in a family N of topological groups is cancellable. In particular, the additive group R of reals and Q of rationals endowed with the usual topologies are both cancellable. We also show that every topological group in a family M of topological groups is cancellable. In particular, every topologically simple non-abelian group is cancellable. It is also proved that every topologically simple Hopfian or co-Hopfian topological group is cancellable.

Let R L be the ring of continuous real-valued functions on a completely regular frame L. We study the class of prime ideals P of the ring R L determined by the condition: R L / P is a real-closed ring. We give some necessary and sufficient frame-theoretic conditions for an ordered integral domain of the form R L / P to be a real-closed ring, when P is a prime z-ideal (d-ideal) of R L . A completely regular frame L is called an SV-frame if for every prime ideal P of R L , the ordered integral domain R L / P is a real-closed ring. It is shown that every C ⁎ -quotient of an SV-frame is an SV-frame. We also show that open quotients ↓c in an SV-frame are SV-frames for all cozero elements c. Larson [22] has given a topological characterization of compact SV-spaces. By extending this characterization to frames, we show that the compactness limitation can really be relaxed, even in spaces, and so strengthen Larson's result.

Suppose M = V ∪ S W is a Heegaard splitting of M where V contains the only one essential disk D V in V and d ( S ) = n . Then, D V is separating and cuts V into two trivial compression bodies V 1 and V 2 . Suppose F 1 is a component of ∂ − V such that F 1 = ∂ − V 1 and S 1 = ∂ + V 1 ∩ S . Then, we can define ψ F 1 : C ( S ) → C ( F 1 ) . Suppose X is a full simplex of C ( F 1 ) , H X is a handlebody obtained by attaching 2-handles to F 1 along X and capping off possible 2-spheres by 3-handles. We denote V ∪ H X by V H . Then, M H = V H ∪ S H W is a Heegaard splitting of M H and M H is said to be a handlebody filling of M. In this paper, we suppose that 0 d ( S H ) n , then there is a constant M such that either if 0 k ≤ M 2 and 2 k − 2 ≤ d C ( F 1 ) ( D ( H X ) , ψ F 1 ( D W ) ) ≤ 2 k − 1 , then d ( S H ) = k or M 2 d ( S H ) n .

We investigate ideal versions of Scheepers' S 1 ( Γ , Γ ) -space, Arkhangel'skiĭ's α 4 -space and Scheepers' monotonic sequence selection property, i.e., S 1 ( I - Γ , J - Γ ) -space, S 1 ( I - Γ 0 , J - Γ 0 ) -space and S 1 ( I - Γ 0 m , J - Γ 0 ) -space, respectively. We show that cardinal invariant λ ( I , J ) introduced in [42] is their common critical cardinality and we study this combinatorial characteristic in its own. For instance, we show that min ⁡ { cov ⁎ ( I ) , b } ≤ λ ( I , J ) ≤ b J and consequently if cov ⁎ ( I ) ≥ b and J has the Baire property then λ ( I , J ) = b . Moreover, we show that control sequence 〈 2 − n : n ∈ ω 〉 in the definition of ( I , s J ) wQN -space in [6] is not essential.

A space X is said to be Menger if for any sequence { U n : n ∈ ω } of open covers of X, there exist finite V n ⊂ U n ( n ∈ ω ) such that ⋃ { V n : n ∈ ω } is a cover of X. For a zero-dimensional space X, let C p ( X , 2 ) be the space of all { 0 , 1 } -valued continuous functions with the topology of pointwise convergence. Bernal-Santos and Tamariz-Mascarua [4] gave several results when C p ( X , 2 ) is Menger. In this paper, we give some improvements of them.

O , Ω and Γ denotes the family of all open, open ω- and open γ-covers of a topological space X, respectively. O s h , Ω s h and Γ s h denotes the corresponding families of shrinkable covers. Let Ψ be one of the symbols O , Ω or Γ. We introduce a property ( Ψ 0 ) of a set of real functions on X. Ψ 0 ( F ) is the set of all subsets of a family of real functions F possessing the property ( Ψ 0 ) . Now, let Φ be one of symbols Ω or Γ. Then for any couple 〈 Φ , Ψ 〉 different from 〈 Ω , O 〉 , a normal topological space X is an S 1 ( Φ s h , Ψ ) -space, if and only if the set C p ( X ) satisfies S 1 ( Φ 0 , Ψ 0 ) , i.e., if S 1 ( Φ 0 ( C p ( X ) ) , Ψ 0 ( C p ( X ) ) ) holds true. Similarly, X is an S 1 ( Φ , Ψ ) -space if and only if the set of non-negative upper semicontinuous real functions on X satisfies S 1 ( Φ 0 , Ψ 0 ) .

Todorcevic introduced the forcing axiom PFA(S) and established many consequences. We contribute to this project. In particular, we consider status under PFA(S) of two important consequences of PFA concerning spaces of countable tightness. We prove that the existence of a Souslin tree does not imply the existence of a compact non-sequential space of countable tightness. We contrast this with M.E. Rudin's result that the existence of a Souslin tree does imply the existence of an S-space (and the later improvement by Dahrough to a compact S-space). On the other hand, PFA(S) implies there is a first-countable perfect pre-image of ω 1 that contains no copies of ω 1 .

In this paper we use some cardinal functions introduced in [3] , [5] and [1] , and the technique of elementary submodels to generalize some cardinal inequalities. Also, we introduce the cardinal functions, denoted by ( γ , θ ) − w L θ and U θ ⁎ , and we prove that: | X | ≤ 2 ( γ , θ ) − w L θ ( X ) χ ( X ) , for any T 1 -space X with U θ ⁎ ( X ) ≤ ω .

In this note we present some results on typical properties of the chain recurrent sets of bounded Baire one functions. In particular we show that typical members of bounded Baire one functions have closed chain recurrent sets and μ ( A ( f ) ) = 0 , where μ is an arbitrary finite Borel measure on the unit interval and A ( f ) is the closure of the set of chain recurrent points of f or any of its subsets.

We prove that every cellular-Lindelof space is weakly linearly Lindelof and hence monotonically normal cellular-Lindelof spaces must be Lindelof. We also show that there exists a cellular-Lindelof locally Lindelof P-space that is not weakly Lindelof. Under CH, we establish that every first countable normal weakly linearly Lindelof space is weakly Lindelof and has cardinality not exceeding c . Our results solve an open question published by Bella and Spadaro.

We investigate basic properties of three cardinal invariants involving ω 1 : stick, antichain number, and matching number. The antichain number is the least cardinal κ for which there does not exist a subcollection of size κ of uncountable subsets of ω 1 with pairwise finite intersections and the matching number is the least cardinal κ for which there exists a subcollection X of size κ of order-type ω subsets of ω 1 so that every uncountable subset of ω 1 has infinite intersection with a member of X. We demonstrate how these numbers are affected by Cohen forcing and also prove some results about the effect of Hechler forcing. We also introduce a forcing notion to increase the matching number, and study its basic properties.

We prove that for each natural number m ≥ 5 , there exists a Tychonoff Moore space with a diagonal of rank exactly m and a zero-set diagonal. We also construct a non-submetrizable Tychonoff Moore space with a diagonal of infinite rank and a zero-set diagonal.

Given compact metric spaces X 1 and X 2 , closed subsets Y 1 and Y 2 of X 1 and X 2 respectively, and an upper semicontinuous function g : Y 1 ⊸ Y 2 , we define a special kind of extension f : X 1 ⊸ X 2 of the function g and prove some of its properties. Using this special extension, we answer a question stated by Banic, Crepnjak, Merhar and Milutinovic. We also introduce the notions of the weak L -extension property and the strong L -extension property of an inverse sequence for any property L of graphs of set-valued functions. We show that, in the case where L means the surjectivity of such graphs, not all inverse sequences of compact metric spaces have the strong L -extension property but they do have the weak L -extension property.

In this paper, we make some observations on chain conditions and dually properties. In particular, we show that: (1) A subspace X ⊂ ω 1 ω is dually CCC then e ( X ) ≤ ω and a normal subspace X ⊂ ω 1 ω is DCCC if and only if e ( X ) ≤ ω ; (2) There is a Tychonoff pseudocompact subspace X ⊂ ( ω 1 + 1 ) 2 which is not dually CCC; (3) In the class of o-semimetrizable spaces, dually separable is self-dual with respect to neighbourhood assignments. As an application, we obtain an example of a CCC normal Moore space which is not dually separable under MA+¬CH; (4) There exists an example of a large normal CCC semi-stratifiable space, which answers a question of Xuan and Song (2018) [21, Question 4.11] ; (5) Every dually separable and monotonically monolithic space is Lindelof, which gives a partial answer to a question of Alas, Junqueira, van Mill, Tkachuk and Wilson (2011) [2, Question 2.1] ; (6) A dually separable Hausdorff space with a strong rank 1-diagonal has cardinality at most 2 c . The conclusion is also true for regular spaces if we replace “strong rank 1-diagonal” with “ G δ -diagonal”; (7) A dually separable ω-monolithic Hausdorff space with a G δ -diagonal has cardinality at most c .

For a Catalan state C of a lattice crossing L ( m , n ) with no returns on one side, we find its coefficient C ( A ) in the Relative Kauffman Bracket Skein Module expansion of L ( m , n ) . We show, in particular, that C ( A ) can be found using the plucking polynomial of a rooted tree with a delay function associated to C. Furthermore, for C with returns on one side only, we prove that C ( A ) is a product of Gaussian polynomials, and its coefficients form a unimodal sequence.

Given a topological property (or a class) P, the class P ′ consists of spaces X such that for any neighbourhood assignment ϕ on X, there exists a subspace Y ⊂ X with property P for which ϕ ( Y ) = ⋃ { ϕ ( y ) : y ∈ Y } is dense in X. The class P ′ are called the weak dual of P or weakly dually P (with respect to neighbourhood assignments). We establish that DCCC is weakly self-dual in the class of weakly regular spaces. If P ∈ { weakly Lindelof , CCC , separable } , then P is weakly self-dual in the class of Baire developable spaces. By using Erdos–Rado's theorem, we also prove that: (1) If X is a Baire, weakly dually CCC Hausdorff space with a rank 2-diagonal, then X has cardinality at most 2 ω ; (2) If X is a Baire, weakly dually DCCC Hausdorff space with a rank 3-diagonal, then X has cardinality at most 2 ω ; (3) If X is a weakly dually DCCC Hausdorff space with a rank 4-diagonal, then X has cardinality at most 2 ω .

Whenever P is a topological property, we say that a topological space X is star P if whenever U is an open cover of X, there is a subspace Y ⊂ X with property P such that St ( Y , U ) = X . We study the relationships of star P properties for P ∈ { DCCC , weakly Lindelof , CCC } with other chain conditions. By using Erdos–Rado's theorem, we also establish several cardinal inequalities for star CCC spaces.

Made available in DSpace on 2019-10-06T16:57:25Z (GMT). No. of bitstreams: 0 Previous issue date: 2019-02-15 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Let G be a group, W a G-set with [G:Gw]=∞ for all w∈W, where Gw denotes the point stabilizer of w∈W. Considering the restriction map resW G:H1(G,Z2G)→∏w∈EH1(Gw,Z2G), where E is a set of orbit representatives for the G-action in W, we define an algebraic invariant denoted by E‾(G,W). In this paper, by using the relation of this invariant with the end e(G) defined by Freudenthal–Hopf–Specker and a Swarup's Theorem about splittings of groups adapted to a family of subgroups, we show, for G finitely generated and W a G-set which falls into many finitely G-orbits, that (G,W) is adapted if, and only if, E‾(G,W)≥2. IBILCE - UNESP - São Paulo State University, Rua Cristovão Colombo, 2265 IBILCE - UNESP - São Paulo State University, Rua Cristovão Colombo, 2265 FAPESP: 2012/24454-8