Rough weighted 𝓘-limit points and weighted 𝓘-cluster points in θ-metric space

  • 1 Department of Mathematics, University of North Bengal, Rajarammohunpur, Darjeeling-734013, West Bengal, India
  • 2 Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India
Sanjoy Ghosal
  • Corresponding author
  • Department of Mathematics, University of North Bengal, Rajarammohunpur, Darjeeling-734013, West Bengal, India
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and Avishek Ghosh

Abstract

In 2018, Das et al. [Characterization of rough weighted statistical statistical limit set, Math. Slovaca 68(4) (2018), 881–896] (or, Ghosal et al. [Effects on rough 𝓘-lacunary statistical convergence to induce the weighted sequence, Filomat 32(10) (2018), 3557–3568]) established the result: The diameter of rough weighted statistical limit set (or, rough weighted 𝓘-lacunary limit set) of a sequence x = {xn}n∈ℕ is 2rlim infnAtn if the weighted sequence {tn}n∈ℕ is statistically bounded (or, self weighted 𝓘-lacunary statistically bounded), where A = {k ∈ ℕ : tk < M} and M is a positive real number such that natural density (or, self weighted 𝓘-lacunary density) of A is 1 respectively. Generally this set has no smaller bound other than 2rlim infnAtn. We concentrate on investigation that whether in a θ-metric space above mentioned result is satisfied for rough weighted 𝓘-limit set or not? Answer is no. In this paper we establish infinite as well as unbounded θ-metric space (which has not been done so far) by utilizing some non-trivial examples. In addition we introduce and investigate some problems concerning the sets of rough weighted 𝓘-limit points and weighted 𝓘-cluster points in θ-metric space and formalize how these sets could deviate from the existing basic results.

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