nLab thread

A thread is an element of a cofiltered limit of topological spaces (usually studied in the generality of projective spectra?) or (in a more rarely used terminology) of a cofiltered limit of sets.

Thus let $F: D^{op}\to Set$ be a functor where $D$ is a small filtered category (for example, a directed set). Then $lim F = lim_{d\in D} F(d)$ consists of families $(s_d)_{d\in D}$ where $s_d \in F(d)$ and for every morphism $\delta:d\to e$ in $D$, $F(\delta)(s_e) = s_d$. Such families are called threads.

If $F: D^{op}\to Top$ is a functor where $D$ is a small filtered category then $lim F$ has the same underlying set (of threads) as the composition $U\circ F$ where $U:Top\to Set$ is the forgetful functor; the topology of $lim F$ is the subspace topology on $lim (U\circ F)$ understood as a subset of the Cartesian product $\prod_d F(d)$ equipped with the (product)Tihonov's topology.