Abstract
The concept of bipolar fuzziness is of relatively recent origin where in addition to the presence of a property, which is done in fuzzy theory, the presence of its counter-property is also taken into consideration. This seems to be much natural and realistic. In this paper, an attempt has been made to incorporate this bipolar fuzziness in topological perspective. This is done by introducing a notion of bipolar gradation of openness and to redefine the bipolar fuzzy topology. Furthermore, a notion of bipolar gradation preserving map is given. A concept of bipolar fuzzy closure operator is also introduced and its characteristic properties are studied. A decomposition theorem involving our bipolar gradation of openness and Chang type bipolar fuzzy topology is established. Finally, some categorical results of bipolar fuzzy topology (both Chang type and in our sense) are proved.
Highlights
From the very beginning of the invention of fuzzy sets by Zadeh [1], many authors have contributed towards fuzzifying the topological concept
The definition of a Bipolar fuzzy topology (BF T) S introduced in [25] looks similar to the definition of Chang fuzzy topological space where the bipolar fuzzy open sets are considered as a crisp collection over some universe
We introduce a definition of bipolar gradation of openness of bipolar fuzzy subsets of X and give a new definition of bipolar fuzzy topological spaces
Summary
From the very beginning of the invention of fuzzy sets by Zadeh [1], many authors have contributed towards fuzzifying the topological concept. A crisp collection never looked good for a proper justification for fuzzifying the topological concept This absence of fuzziness in Chang fuzzy topology was pointed out by Sostak [7], Ying [8], Chattopadhyay et al [9], Gregoroi [10], and Mondal [11]. The definition of a BF T S introduced in [25] looks similar to the definition of Chang fuzzy topological space where the bipolar fuzzy open sets are considered as a crisp collection over some universe. This looks to be a drawback in proper bifuzzification of the topological concept. It is shown that the bipolar fuzzy topologies in our sense and the bipolar gradation preserving mapping is a topological category
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