Search results for "Extension topology"

showing 4 items of 4 documents

Generalized fuzzy topology versus non-commutative topology

2011

The paper introduces a modification of the notions of generalized fuzzy topological space of Demirci and quantal space of Mulvey and Pelletier, suitable to explore interrelations between point-set lattice-theoretic topology and non-commutative topology developed in the framework of C^*-algebras or (more recently) of quantales. As a consequence of the new approach, a generalization of the concept of topological system of Vickers arises. Moreover, the currently dominating variable-basis topological setting in the fuzzy community, due to Rodabaugh, appears to be ''fixed-basis''.

Weak topologyArtificial IntelligenceLogicTrivial topologyExtension topologyProduct topologyInitial topologyGeneral topologyParticular point topologyTopological spaceTopologyMathematicsFuzzy Sets and Systems
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Composite variety-based topological theories

2012

Motivated by the recent result of Rodabaugh on categorical redundancy of lattice-valued bitopology, the paper considers another viewpoint on the topic, based on the notion of composite variety-based topological theory. The new concept, apart from providing a variable-basis generalization of bitopology, incorporates the most important approaches to topology currently developed in the fuzzy community, bringing forward their categorically algebraic properties, which are cleared from point-set lattice-theoretic dependencies. Dwelling on different ways of interaction between composite topology and topology, e.g., embedding the former into the latter as a full bicoreflective subcategory, we final…

Weak topologyArtificial IntelligenceLogicSubbaseMathematics::General TopologyExtension topologyProduct topologyInitial topologyGeneral topologyTopological spaceParticular point topologyTopologyMathematicsFuzzy Sets and Systems
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Categorically algebraic topology versus universal topology

2013

This paper continues to develop the theory of categorically algebraic (catalg) topology, introduced as a common framework for the majority of the existing many-valued topological settings, to provide convenient means of interaction between different approaches. Motivated by the results of universal topology of H. Herrlich, we show that a concrete category is fibre-small and topological if and only if it is concretely isomorphic to a subcategory of a category of catalg topological structures, which is definable by topological co-axioms.

Artificial IntelligenceLogicMathematics::Category TheoryCategory of topological spacesAlgebraic topology (object)Extension topologyTopological groupGeneral topologyInitial topologyTopological spaceParticular point topologyTopologyMathematicsFuzzy Sets and Systems
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Fuzzy algebras as a framework for fuzzy topology

2011

The paper introduces a variety-based version of the notion of the (L,M)-fuzzy topological space of Kubiak and Sostak and embeds the respective category into a suitable modification of the category of topological systems of Vickers. The new concepts provide a common framework for different approaches to fuzzy topology and topological systems existing in the literature, paving the way for studying the problem of interweaving algebra and topology in mathematics, which was raised by Denniston, Melton and Rodabaugh in their recent research on variable-basis topological systems over the category of locales.

AlgebraPure mathematicsTopological algebraArtificial IntelligenceLogicCategory of topological spacesExtension topologyInitial topologyGeneral topologyTopological spaceTopological vector spaceHomeomorphismMathematicsFuzzy Sets and Systems
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