On stratified L-convergence spaces: Fischer's diagonal axiom

Li, Lingqiang; Jin, Qiu; Hu, Kai

doi:10.1016/j.fss.2014.09.001

Cited by 29 publications

(7 citation statements)

References 17 publications

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“…The convergence theory associated with SL-Top is presented in [28]. (2) The convergence theory associated with STrL-Top is given in [14].…”

Section: Definition 3 ([14])mentioning

confidence: 99%

“…(1) The Gähler compression operator was presented by Jäger [4]. Then, some types of lattice-valued Fischer diagonal conditions were proposed to discuss the topologicalnesses of different L-convergence spaces [13,18,28]. Particularly, it was proved in [28] that stratified L-convergence space with the considered Fischer diagonal condition precisely characterizes stratified L-topological space.…”

Section: Introductionmentioning

confidence: 99%

“…Then, some types of lattice-valued Fischer diagonal conditions were proposed to discuss the topologicalnesses of different L-convergence spaces [13,18,28]. Particularly, it was proved in [28] that stratified L-convergence space with the considered Fischer diagonal condition precisely characterizes stratified L-topological space. Meanwhile, some lattice-valued dual Fischer diagonal conditions were introduced to study the regularities of different L-convergence spaces [19,27,31].…”

Section: Introductionmentioning

confidence: 99%

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Fuzzy Counterparts of Fischer Diagonal Condition in ⊤-Convergence Spaces

Jin

Jiang

2019

Mathematics

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Fischer diagonal condition plays an important role in convergence space since it precisely ensures a convergence space to be a topological space. Generally, Fischer diagonal condition can be represented equivalently both by Kowalsky compression operator and Gähler compression operator. ⊤-convergence spaces are fundamental fuzzy extensions of convergence spaces. Quite recently, by extending Gähler compression operator to fuzzy case, Fang and Yue proposed a fuzzy counterpart of Fischer diagonal condition, and proved that ⊤-convergence space with their Fischer diagonal condition just characterizes strong L-topology—a type of fuzzy topology. In this paper, by extending the Kowalsky compression operator, we present a fuzzy counterpart of Fischer diagonal condition, and verify that a ⊤-convergence space with our Fischer diagonal condition precisely characterizes topological generated L-topology—a type of fuzzy topology. Hence, although the crisp Fischer diagonal conditions based on the Kowalsky compression operator and the on Gähler compression operator are equivalent, their fuzzy counterparts are not equivalent since they describe different types of fuzzy topologies. This indicates that the fuzzy topology (convergence) is more complex and varied than the crisp topology (convergence).

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“…The convergence theory associated with SL-Top is presented in [28]. (2) The convergence theory associated with STrL-Top is given in [14].…”

Section: Definition 3 ([14])mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fuzzy Counterparts of Fischer Diagonal Condition in ⊤-Convergence Spaces

Jin

Jiang

2019

Mathematics

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“…In the lattice-valued case of the underlying lattice L=[0,1], Lowen et al [28,29] consider fuzzy convergence spaces as a generalization of Choquent [2] convergence spaces and the resulting category has, among other things, function space, where prime prefilters play a crucial role as Jäger [22] pointed out. Beside prefilter, there are other kinds of lattice-valued filters in literature of fuzzy topology or nontrivial lattice-valued convergence theory, see Eklund and Gähler [3], Fang [5,6,[8][9][10], Flores and others [4], Höhle andŠostak [16], Jäger [21,22,24,25],, Jäger and Burton [23], Li [26,27], Pang [32][33][34], Yao [42] and others. Besides Lowen's function space of fuzzy convergence spaces, there are some significant works of function spaces established by using different lattice-valued filters.…”

Section: Motivation and Introductionmentioning

confidence: 99%

Function spaces based on L-sets

Fang

Yue

2020

Filomat

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For a commutative, integral, and divisible quantale L, a concept of top L-convergence spaces based on L-sets other than crisp sets is proposed by using a kind of L-filters, namely limited L-filters defined in the paper. Our main result is the existence of function spaces in the the concrete category of top L-convergence spaces over the slice category Set#L rather than the category Set of sets, such that the concrete category of top L-convergence spaces over the slice category Set#L is Cartesian closed. In order to support the existence of top L-convergence spaces, some nontrivial examples of limited L-filters and top L-convergence spaces are presented also.

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“…In recent years, two kinds of lattice-valued convergence spaces received much attention: (1) the theory of stratified L-generalized convergence spaces based on L-filters, which is initiated by Jäger [5] and then developed by many researchers [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]; and (2) the theory of -convergence spaces based on -filters, which is investigated by Fang [26] in 2017. The topologicalness in stratified L-generalized convergence spaces was studied by Jäger [27][28][29] and Li [30,31], the p-topologicalness and p-topological modifications in stratified L-generalized convergence spaces were discussed by Li [32,33].…”

Section: Introductionmentioning

confidence: 99%

p-Topologicalness—A Relative Topologicalness in ⊤-Convergence Spaces

Li¹

2019

Mathematics

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In this paper, p-topologicalness (a relative topologicalness) in ⊤-convergence spaces are studied through two equivalent approaches. One approach generalizes the Fischer’s diagonal condition, the other approach extends the Gähler’s neighborhood condition. Then the relationships between p-topologicalness in ⊤-convergence spaces and p-topologicalness in stratified L-generalized convergence spaces are established. Furthermore, the lower and upper p-topological modifications in ⊤-convergence spaces are also defined and discussed. In particular, it is proved that the lower (resp., upper) p-topological modification behaves reasonably well relative to final (resp., initial) structures.

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On stratified L-convergence spaces: Fischer's diagonal axiom

Cited by 29 publications

References 17 publications

Fuzzy Counterparts of Fischer Diagonal Condition in ⊤-Convergence Spaces

Fuzzy Counterparts of Fischer Diagonal Condition in ⊤-Convergence Spaces

Function spaces based on L-sets

p-Topologicalness—A Relative Topologicalness in ⊤-Convergence Spaces

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