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Submission declined on 19 October 2025 by Anerdw (talk).
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| Submission declined on 18 October 2025 by GGOTCC (talk). This submission is not adequately supported by reliable sources. Reliable sources are required so that information can be verified. If you need help with referencing, please see Referencing for beginners and Citing sources. Declined by GGOTCC 26 days ago. | |
Comment: Sources 1 and 2 are not intellectually independent of each other. Source 3 isn't significant coverage of semitopology. Anerdw (talk) 05:27, 19 October 2025 (UTC)
- Comment: Certainly a notable topic, but most of the article is unsourced. GGOTCC 03:02, 18 October 2025 (UTC)
- Comment: In accordance with Wikipedia's Conflict of interest policy, I disclose that I have a conflict of interest regarding the subject of this article. Bellissimogiorno (talk) 19:02, 17 October 2025 (UTC)
Semitopology generalises general topology by removing the condition that the intersection of two open sets must be open.[1]
Definition
A semitopology is a pair (X,Opens) of a set X and a subset Opens ⊆ Powerset(X) such that Opens contains the empty set, X, and is closed under arbitrary sets unions.[1]: Definition 1.1.2
Motivation
Open sets of semitopologies abstract the notion of quorum from distributed computing.
An open set is interpreted as an actionable coalition, this being a set of participants in a distributed system with the power to collaborate to act to advance their local state (i.e. to take a step in some distributed protocol).
Using this interpretation, aspects of distributed algorithms can be framed using topological language. Notably:
- The notion of topological continuity can be identified with consensus.[1]: Lemma 2.2.4
- The characteristic quorum intersection property that any two quorums intersect,[2]: 2.7.3 Quorums can be identified with the negation of the T2 Hausdorff separation property that any two distinct points have non-intersecting open neighbourhoods.[1]: Remark 3.6.10
Examples
- Every topology is trivially a semitopology.
- Let X be a finite set with n elements. Write MoreThan(k,X) for the set of subsets of X that contain more than k elements. Then (X,MoreThan(k,X)) is a semitopology. This semitopology is not in general a topology.
References
- 1 2 3 4 Gabbay, Murdoch (20 January 2025). "Semitopology: a topological approach to decentralized collaborative action". Journal of Logic and Computation. 35 (5) exae050. doi:10.1093/logcom/exae050.
- ↑ Cachin, Christian; Guerraoui, Rachid; Rodrigues, Luı́s (2011). Introduction to Reliable and Secure Distributed Systems (2 ed.). Springer. ISBN 978-3-642-15259-7.
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