  We introduce semitopologies, a generalisation of point-set topology that
removes the restriction that intersections of open sets need necessarily be
open.
  The intuition is that points are participants in some distributed system, and
an open set is a collection of participants that can collaborate to update
their local state by taking a distributed collaborative action; we call this an
actionable coalition. What constitutes an actionable coalition depends on what
actions we want to model. Intuitive examples include 'a group of people that is
collectively strong enough to lift a rock', where the state update is very
simply 'holding rock low' to 'holding rock high' and this update is common to
all participants in the actionable coalition. Or, consider 'two people wishing
to barter a can of juice for a bar of chocolate', in which case the coalition
is any such pair and the state updates differ between participants to flip them
between 'has/has no juice' and 'has/has no chocolate'. A characteristic of
these systems is that state updates are local to the coalition, voluntary, may
vary between participants, and are not assumed subject to permission or
synchronisation by a central authority. Peer-to-peer computer networks,
including filesharing and blockchain systems, provide motivating examples from
computing.
  This monograph presents a comprehensive view of semitopologies which includes
point-set semitopology, algebra, and logic inspired by these considerations.
This is interesting in and of itself and it provides a conceptual framework
within which to understand a useful class of distributed systems.
