• Raghavendra K
• Veeresha A Sajjanara
• Govardhanareddy H G

This paper examines R#-closed sets, a class of generalized closed sets in topology situated between g-closed and rg-closed sets, and proposes a framework for instantiating them within digital (Khalimsky) topological spaces used to model pixel grids and other discrete data domains. After reviewing the hierarchy of generalized closed sets and the foundations of digital topology, the chapter formally defines digital R#-closed sets, proves that the closure-under-union property of R#-closed sets carries over unchanged to the digital setting, and establishes a parity condition under which rectangular pixel blocks are guaranteed to be R#-closed. Building on these results, the chapter proposes R#-closed regions as a tunable middle ground between the strict requirements of digital g-closed sets and the permissive requirements of digital rg-closed sets, with potential relevance to block-based image partitioning, boundary-stability analysis, and discrete data topology more broadly. The chapter closes by identifying open problems for empirical and algorithmic follow-up work in computer science and engineering.

R#-closed sets; generalized closed sets; g-closed sets; rg-closed sets; digital topology; Khalimsky space; image processing; boundary-stability analysis; Alexandrov spaces; topological data analysis

"R#-Closed Sets in Digital Topology Toward Noise-Tolerant Region ", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.13, Issue 10, page no.a443-a452, October-2026, Available :http://www.jetir.org/papers/JETIR2610047.pdf

"R#-Closed Sets in Digital Topology Toward Noise-Tolerant Region ", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.13, Issue 10, page no. ppa443-a452, October-2026, Available at : http://www.jetir.org/papers/JETIR2610047.pdf