Abstract
For the first time we introduce non-standard neutrosophic topology on the extended non-standard analysis space, called non-standard real monad space, which is closed under neutrosophic non-standard infimum and supremum. Many classical topological concepts are extended to the non-standard neutrosophic topology, several theorems and properties about them are proven, and many examples are presented.
Highlights
Smarandache [4,5] introduced at the beginning of 2019 for the first time, the left monad closed to the right, the right monad closed to the left, and unpierced binad, defined as below: Left Monad Closed to the Right
“m” written above the standard real number “a” means: a standard real number (0, or nothing above), or a left monad (− ), or a left monad closed to the right (−0 ), or a right monad (+ ), or a right monad closed to the left (0+ ), or a pierced binad (−+ ), or a unpierced binad (−0+ ) respectively
We have introduced for the first time the non-standard neutrosophic topology, non-standard neutrosophic toplogical space and subspace constructed on the non-standard unit interval]−0, 1+[M that is formed by real numbers and positive infinitesimals and open monads, together with several concepts related to them, such as: non-standard neutrosophic open/closed sets, non-standard neutrosophic closure and interior of a given set, and non-standard neutrosophic product topology
Summary
The purpose of this study is to initiate for the first time a new field of research, called non-standard neutrosophic algebraic structures, and we start with non-standard neutrosophic topology (NNT) in this paper. Being constructed on the set of hyperreals, that includes the infinitesimals, NNT can further be utilized in neutrosophic calculus applications. As a branch of mathematical logic, non-standard analysis [1] deals with hyperreal numbers, which include infinitesimals and infinities. In 1998, Smarandache [3] used non-standard analysis in philosophy and in neutrosophic logic, in order to differentiate between absolute truth (which is truth in all possible worlds, according to Leibniz), and relative truth (which is, according to the same Leibniz, truth in at least one world). This is analogously for absolute falsehood vs relative falsehood, and absolute indeterminacy vs relative indeterminacy He extended [3] the use of non-standard analysis to neutrosophic set By R+ * we denote the set of positive non-zero hyperreal numbers. Μ (a) is a monad (halo) of an element a ∈ R*, which is formed by a subset of numbers infinitesimally close (to the left-hand side, or right-hand side) to a
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