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 $F:D^{\mathrm{op}}\to \mathrm{Set}$ 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 $F:D^{\mathrm{op}}\to \mathrm{Top}$ 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 $U:\mathrm{Top}\to \mathrm{Set}$ 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.