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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

In this paper we define associated spherical partner, radial function and radial spherical function for space curve which doesn't lie on the sphere in Euclidean 3-space. Using these notions we discuss some special curves and their associated spherical partners. Especially we give a new necessary and sufficient condition for the rectifying curves in Euclidean 3-space.

The present paper studies two topologies of the quotient spaces related to the Alexandroff one point compactification of the Marcus-Wyse (M-, for brevity) topological plane. The topologies are exactly proved to be the cofinite particular point and the excluded two points topology. We also concern separation axioms and nowhere dense subset properties of the quotient spaces of the M-topological plane. Furthermore, we investigate both a particular point and an excluded point topological structure associated with the quotient spaces of the M-topological plane. Finally, we investigate the fixed point property of the compactification of the M-topological plane. Unlike the non-fixed point property of the Hausdorff compactification of the Euclidean topological plane, we prove that every continuous self-bijection of the compactification of the M-topological plane has a fixed point.

A G-map f : X → Y is called isovariant if it preserves isotropy groups (stabilizers), that is, G x = G f ( x ) for all x ∈ X . For a given compact group G we consider the category I s o v - M of metrizable G-spaces and isovariant maps. In a natural way the notion of an isovariant absolute (neighborhood) extensor ( I s o v - A ( N ) E ) can be introduced; this was done in [1] . Using this notion we define the concept of an isovariant fibrant space and study its properties. In particular, we show that every compact metric G-space admits an isovariant fibrant extension.

We give a classification of Levi-umbilical real hypersurfaces, whose Levi form is proportional to the induced metric, in the complex two plane Grassmannian G 2 ( C m + 2 ) , the complex quadric Q n , or their noncompact dual spaces.

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.

We present a necessary and sufficient condition for metrizability of a regular space X covered by an open covering each of its elements is metrizable by the restriction of a single metric d on X .

A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation generated by flat version of classical Reidemeister moves and virtual Reidemeister moves such that Reidemeister moves of type 3 are forbidden. In this paper we discuss colorings of virtual diagrams using an algebra, called a doodle switch, and define an invariant of virtual doodles. Besides usual colorings of diagrams, we also introduce doubled colorings.

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.

We show a method to determine topological complexity from the fibrewise view point, which provides an alternative proof for tc ( K ) = 4 , where K denotes Klein bottle.

In this paper, we study when a strongly freely decomposable mapping is almost monotone. Also, we define the notion of i-unicoherent continuum and present some properties and examples of this class of continua.

We introduce a family of polynomial invariants for flat virtual knots which extend the polynomial introduced by Kauffman and Richter, and we give several properties and examples.

It is shown that there are infinitely many immersed 2-knots with more than any previously given number of double point singularities which are not equivalent to the connected sum of any immersed 2-knot and any unknotted immersed sphere.

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.

Recall that a space X is selectively pseudocompact if for every sequence { U n : n ∈ N } of non-empty open subsets of X one can choose a point x n ∈ U n for all n ∈ N such that the resulting sequence { x n : n ∈ N } has an accumulation point in X. This notion was introduced under the name strong pseudocompactness by Garcia-Ferreira and Ortiz-Castillo; the present name is due to Dorantes-Aldama and the first listed author. In 2015, Garcia-Ferreira and Tomita constructed a pseudocompact Boolean group that is not selectively pseudocompact. We prove that if the subgroup topology on every countable subgroup H of an infinite Boolean topological group G is finer than its maximal precompact topology (the so-called Bohr topology of H), then G is not selectively pseudocompact, and from this result we deduce that many known examples in the literature of pseudocompact Boolean groups automatically fail to be selectively pseudocompact. We also show that, under the Singular Cardinal Hypothesis, every infinite pseudocompact Boolean group admits a pseudocompact reflexive group topology which is not selectively pseudocompact.

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 first obtain a sort of converse of Cubas's result [4] : If a Lipschitz continuous map f can be C 0 Gromov-Hausdorff approximated by periodic orbits and the convergence rate is faster than that of the corresponding periods, then f is transitive. This conclusion is false without such a convergence rate requirement. Next we prove that the set of periodic orbits is dense in the set of chain transitive maps with respect to the C 0 Gromov-Hausdorff distance.

A 4D universe is a 4-dimensional boundary-less connected oriented manifold with every closed 3-manifold (i.e., a 3-dimensional closed connected oriented manifold) embedded. A 4D punctured universe is a 4-dimensional boundary-less connected oriented manifold with the punctured manifold of every closed 3-manifold embedded. Every 4D universe and every 4D punctured universe are open 4-dimensional manifolds. If a closed 3-manifold is considered as a 3D universe, then every 4D spacetime is embedded in every 4D universe and hence every 4D universe is a classifying space for every spacetime. In this paper, it is observed that a full 4D universe is produced by collision modifications between 3-sphere fibers in the 4D spherical shell (i.e., the 3-sphere bundle over the real line) embedded properly in any 5-dimensional open manifold. As a previous result, it was shown that any 4D universe and 4D punctured universe must have infinity on some homological indexes. It is shown in this paper that the second rational homology groups of every 4D universe and every 4D punctured universe are always infinitely generated.

High-dimensional networks are a topological generalization of electrical networks. As combinatorial invariants of electrical networks often require enumeration of trees, their high-dimensional analogues are also based on torsion-weighted enumeration of acyclic subcomplexes over the reals. In this paper, for a high-dimensional network with a current generator, we express a current vector and a voltage vector in terms of high-dimensional forests. To this end, we introduce a new simple method, an acyclization in codimension 1, for computing forest-numbers.

The free topological vector space V ( X ) over a Tychonoff space X is a pair consisting of a topological vector space V ( X ) and a continuous mapping i = i X : X → V ( X ) such that every continuous mapping f from X to a topological vector space E gives rise to a unique continuous linear operator f ‾ : V ( X ) → E with f = f ‾ ∘ i . In this paper, the k-property and countable tightness of free topological vector space over some generalized metric spaces are studied. We mainly discuss the characterizations of a space X such that V ( X ) or the fourth level of V ( X ) is a k-space or is of countable tightness, respectively.

We introduce Markov-like functions on finite graphs and define the notation of the same pattern between those Markov-like functions. Then we show that two generalized inverse limits with Markov-like bonding functions on finite graphs having the same pattern are homeomorphic. This result gives a generalization of our recent result [6] .

For a Lie group G, we prove the existence of a universal G-space in the class of all paracompact (respectively, metrizable, and separable metrizable) free proper G-spaces which have a paracompact orbit space. Our approach is based on Milnor's construction E G . We show that E G and the universal free G-spaces in question are G-AE's. Besides, we show how to extend these results to the case of arbitrary metrizable, locally compact, almost connected group actions.

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.

Recently, Zhao and Ho introduced and studied SI-continuous spaces, which can be seen as topological counterparts of continuous posets. The main purpose of this paper is to investigate the relationships between SI-continuous spaces and continuous posets. We prove that a C-space is an SI-continuous space if and only if it is a continuous poset under the specialization order. Furthermore, we introduce the notion of strong SI-continuous spaces and construct an adjunction between the category of domains and the category of strong SI-continuous spaces.

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] .

In this paper, we characterize non-hyperbolic 3-component links in the 3-sphere whose exteriors contain essential 3-punctured spheres with non-integral boundary slopes. We also show the existence of embeddings of some multibranched surfaces in the 3-sphere which satisfy some homological conditions to be embedded in the 3-sphere., 11 pages, 7 figures

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.

Lattice knot statistics which is the study of knotted polygons in the simple cubic lattice have been deeply studied. In this paper, we discuss the enumeration of lattice links and fully-packed lattice links confined to slabs of width 1, as a model for multi-component ring polymers in physics. Moreover, the generating functions for such 1-slab lattice links with respect to the total length and the number of sticks are determined. We also analyze the asymptotic behavior of the growth rates of their cardinality.

[EN] In this paper we show that, in contrast to what happens in a normed space, the weak topology of a finite dimensional asymmetric normed space is not necessarily the same as the topology of the asymmetric norm. We provide a class of finite dimensional asymmetric normed spaces where both topologies coincide. We also prove that the weak topology of an infinite dimensional asymmetric normed space is strictly coarser than the topology of the asymmetric norm. (C) 2019 Elsevier B.V. All rights reserved.

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

This is a sequel of the author's work which aims at extending Jorgensen's work on the once punctured torus to a torus with a cone point. This paper is devoted to a study of cone hyperbolic manifolds with “thin” holonomy representations. We give an explicit construction of such cone manifolds and the “convex cores” of such cone manifolds. We also give two natural parametrizations of such cone manifolds.

We work in set theory without the Axiom of Choice ( AC ) and establish the following results: 1. “Products of ultracompact spaces are ultracompact” + “there exists a free ultrafilter on ω” ( UF ( ω ) ) is equivalent to AC in ZFA ; 2. “Products of ultracompact spaces are ultracompact” does not imply AC in ZF ; 3. AC is equivalent to each of “products of ultracompact spaces are countably compact”, “products of ultracompact spaces are sequentially limit point compact”, “products of ℵ 0 -bounded spaces are countably compact”, “products of ℵ 0 -bounded spaces are sequentially limit point compact”, and “products of ℵ 0 -bounded spaces are ℵ 0 -bounded” in ZFA ; 4. UF ( ω ) is equivalent to each of “every ultracompact space is sequentially limit point compact” and “every ultracompact T 2 space is sequentially limit point compact”; 5. The Axiom of Countable Choice ( AC ω ) is equivalent to “every ℵ 0 -bounded space is countably compact”; 6. AC ω + UF ( ω ) is equivalent to each of “every ultracompact space is countably compact”, “products of countably many ultracompact spaces are countably compact”, “products of countably many ultracompact spaces are sequentially limit point compact”, and “products of countably many ℵ 0 -bounded spaces are ℵ 0 -bounded”; 7. Each of “a T 3 space is ultracompact, if and only if, it is ℵ 0 -bounded” and “a Tychonoff space is ultracompact, if and only if, it is ℵ 0 -bounded” is equivalent to “every filter on ω can be extended to an ultrafilter on ω” ( BPI ( ω ) ), and thus each of the above statements is equivalent to the statement “the Stone space βω of all ultrafilters on ω is compact”. Thus we provide the exact characterizations of the above topological results, the former by Vaughan in “Countably compact and sequentially compact spaces”, and the latter by Bernstein in “A new kind of compactness for topological spaces”, as a specific renowned weak choice principle (which is strictly weaker than the Boolean Prime Ideal Theorem in ZF ); 8. AC ω implies Ginsburg's theorem in “Some results on the countable compactness and pseudocompactness of hyperspaces”, namely “a T 2 space X is ultracompact, if and only if, the hyperspace C L ( X ) (the set of non-empty closed subsets of X with the Vietoris topology) is ultracompact”, which in conjunction with UF ( ω ) implies the weak choice principle A C fin ω ( AC restricted to countable families of non-empty finite sets).

In [11] , Landweber and Stong proved that if a closed spin manifold M admits a smooth S 1 -action of odd type, then its signature sign ( M ) vanishes. In this paper, we extend the result to a torus action of generalized odd type on a closed oriented manifold.

We find a probability measure which counts the pairs of closed commuting subgroups in infinite groups. This measure turns out to be an extension of what was known in the finite case as subgroup commutativity degree. The extremal case of probability one describes the so-called topologically quasihamiltonian groups and is a useful tool in describing the distance of a profinite group from this special class. We have been inspired by an idea of Heyer in the context of our problem.

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.

In 1987, Dress and Scharlau studied the well-known concept of gated sets in metric spaces in order to investigate projections in Tits buildings. In this article, we introduce analogously the concept gated sets in a T 0 -quasi-metric space that we call in-gated set. It turns out that in a T 0 -quasi-metric space, there is a dual concept which we call outgated set. We use these concepts to extend some classical results on gluing a family of hyperconvex metric spaces along a set such that the resulting space preserves hyperconvexity from a metric point of view to quasi-metric settings.

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.

Each partial order generates a transitive quasi-uniformity. In this article we will study the properties of quasi-uniformities that are defined by a partial order and are QH-singular.

For any locally compact abelian periodic group A its automorphism group contains as a subgroup those automorphisms that leave invariant every closed subgroup of A, to be denoted by SAut(A). This subgroup is again a locally compact abelian periodic group in its natural topology and hence allows a decomposition into its p-primary subgroups for p the primes for which topological p-elements in this automorphism subgroup exist. The interplay between the p-primary decomposition of SAut(A) and A can be encoded in a bipartite graph, the mastergraph of A. Properties and applications of this concept are discussed., Oberwolfach Preprints;2018,05

In this paper, a collection of completeness type properties of topological spaces is introduced. They are determined by families of open covers. It is shown that some of the completeness type properties imply isocompactness, and that some of them are productive. This allows to specify new cases when closed pseudocompact subsets of topological spaces are compact. The completeness type properties are also considered below in connection with group extensions of topological groups.

In this paper we modify the definition of Whitney preserving map to introduce a new family of functions; weakly Whitney preserving maps, and we study the relationships of this new family with other families of functions.

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.

In this paper, we introduce sheaves on diffeological spaces as sheaves on the site of plots. We also define quasi-sheaves by presheaves that respect the limit over covering generating families. It is shown that every sheaf on diffeological spaces is a quasi-sheaf. Moreover, every sheaf on a diffeological space gives a sheaf on the D-topological structure. Finally, we characterize a class of sheaves, so-called comma sheaves, and prove that it is equivalent to the comma category of diffeological spaces.

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.

Recently Bernardes and Darji provided a very nice characterization of a residual set of maps of Cantor set in terms of covers of special type. Using their characterization, we provide a more direct description of this class. This way we are able to provide a further characterization of dynamical properties (e.g. shadowing properties, nullness) of maps in the class and further study of what features (e.g. prescribed minimal sets or the values of topological entropy) we can get by small perturbations of a given homeomorphism.

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.