We summarize some facts on chains (totally ordered sets), from an order-theoretic and from a topological point of view. We highlight the fact that many classical theorems that are true for partially ordered sets under some completeness assumption remain true for chains without any kind of completeness.
Topologies of (strong) uniform convergence on bornologies
Toposes of connectivity spaces. Morita equivalences with topological spaces and partially ordered sets in the finite case
Topologies induced by group actions
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