On Pointfree Countably Disconnectivity

Authors

  • Omid Rahmanian Hakim Sabzevari University
  • Prof. Ali Akbar Estaji Hakim Sabzevari Univ.
  • Mostafa Abedi Esfarayen University of Technology

Keywords:

Frame, $F_c$-frame, $c$-basically disconnected, extremally disconnected, Ring of continuous functions with countable images on a frame

Abstract

We introduce pointfree counterparts of countably disconnectivity, specifically countably basically disconnected frames (briefly, $c$-basically disconnected frames) and countably $F$-frames (briefly, $F_c$-frames), along with their frame-theoretic characterizations. We present characterizations of zero-dimensional extremally disconnected frames, zero-dimensional $c$-basically disconnected frames, and zero-dimensional $F_c$-frames $L$ using the ring-theoretic properties of the ring $\mathcal{R}_cL$ of continuous real-valued functions with countable images on $L$. We show that a zero-dimensional frame $L$ is extremally disconnected if and only if any annihilator ideal of $\mathcal{R}_cL$ is generated by an idempotent, $c$-basically disconnected if and only if the annihilator of each element of $\mathcal{R}_cL$ is generated by an idempotent, and an $F_c$-frame if and only if the annihilator of each element of $\mathcal{R}_cL$ is pure.

Published

2025-09-10

Issue

Vol. 19

Section

Vol. 19, No. 5, (2025)

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