Blockchains and peer-to-peer systems represent a paradigm shift toward radically decentralised computation, where systems operate across numerous participants without central control. In the recent paper Decentralised collaborative action: cryptoeconomics in space, the author introduces the mathematical concept of semitopology, in order to model such systems. In the following post, we will break down the core ideas of this paper and its implications for cryptoeconomics.

The main idea here is to model decentralisation mathematically. Traditional distributed systems often rely on centralised components, or it is assumed uniform participant behavior. Contrary to that, decentralised collaborative action systems like blockchains are defined by multi-participant execution, and no central control by any single entity. These two properties are critical to the system's purpose. To model this behavior, the author proposes semitopology, which unlike classical topologies, they relax the requirement that intersections of open sets must be open.

Now, to be more clear and pedagogical: what exactly is a topology? This is a way that mathematicians describe how points in a space relate to each other in terms of "closeness" or "connectedness" but more generally than using precise distances. It focuses on properties that stay the same when you stretch, bend, or twist objects, as long as you don't tear them apart. Let's think of a coffee mug and a donut, which are topologically the same, since ou can stretch and reshape a coffee mug to convert it into a mug, without tearing or gluing parts. Both objects, after every continuous transformation, have only one hole.

More formally, a topology on a set is a collection of subsets called open sets that satisfy three rules:

(a) The empty set and the whole set are always open.
(b) Any union of open sets is open.
(c) Any finite intersection of open sets is open.

An example of a topology would be the following. Consider P = {0,1,2}. A simple topology on P would be the set T = {∅,{0},{0,1},{0,1,2}}, (where ∅ represents an empty set), since

(a) both ∅ and P are in T.
(b) the unions of the elements are also in T, as for instance the union of {0} and {0,1} is {0,1} and it is in T.
(c) the intersections, for instance the intersection of {0} and {0,1} is {0} and is also in T.

For describing a semitopology, consider again P={0,1,2} with T= {∅, {0,1}, {1,2}, {0,1,2}}. This forms a semitopology, since the intersection of {0,1} and {1,2} is {1} and it is not in T, violating the intersection condition.

This simple idea can be related to the concept of actionable coalitions, which are simply subgroups of participants in a group, who can collaborate to update their local state independently of others. In the paper it is defined the concept of cryptoeconomics, which is defined as the study of socioeconomic systems enabled by modern decentralised computer systems. Cryptoeconomics studies incentive mechanisms that align participant behavior with system goals. Therefore, one can connect cryptoeconomics to semitopology by analizing how actionable coalitions enforce consensus and security. For instance, in proof-of-stake blockchains, a majority stake forms an actionable coalition. If two participants' coalitions always intersect, they cannot fork, ensuring agreement. In peer-to-peer networks, the participants form coalitions based on communication links. Overlapping coalitions enable local state updates without global coordination. Finally, semitopology helps verify whether cryptoeconomic rules, such as slashing conditions, ensure coalitions intersect sufficiently to prevent misbehavior.

A system would exhibit decentralised collaborative action if: the participants store and update local state without a global truth source, if they act by choice and not by obligation, if the participants are heterogeneous and may have different goals, resources, or rules; and if the system is permisionless and no central authority grants collaboration rights.

Some of the insights and applications of the concept of semitopology into the decentralised collaborative action are the following. In principle, semitopology can explain why decentralised systems like blockchains avoid forks: intertwined participants' coalitions overlap, forcing an agreement in the systems. For instance, in Bitcoin, miners' majority coalitions intersect, ensuring a single chain. It can be argued too that there are actionable kernels and every semitopology has one, that is a minimal coalition whose decisions propagate to all participants. This replaces a single dictator with a small coalition, raising centralization concerns if the kernel is too small.

As a conclusion, semitopology is offered in this paper as a powerful mathematical tool for analyzing decentralised systems. By treating actionable coalitions as open sets, it is revealed structural properties that hold regardless of specific algorithms. These insights can help engineers to build more robust blockchains by ensuring coalitions intersect, or to improve security in cryptoeconomic mechanisms by formally doing verifications using semitopological axioms. As decentralised systems grow in complexity, semitopology provides a foundational theory to understand and improve their behavior.