# What does “thread” mean in general topology?

I am studying from "R Engelking - General Topology". In p.98 it is written:

Let $\mathbf S=\{X_\sigma,\pi^\sigma_\rho,\Sigma\}$ be an inverse system; an element $\{x_\sigma\}$ of the Cartesian product $\prod_{\sigma\in\Sigma}X_\sigma$ is called a thread of $\bf S$ if $\pi^\sigma_\rho(x_\sigma)=x_\rho$ for any $\sigma,\rho\in\Sigma$ satisfying $\rho\leq\sigma$, and the subspace $\prod_{\sigma\in\Sigma}X_\sigma$ consisting of all threads of $\bf S$ is called the limit of the inverse system $\mathbf S=\{X_\sigma,\pi^\sigma_\rho,\Sigma\}$ and is denoted by $\lim\limits_{\small\leftarrow}\bf S$ or by $\lim\limits_{\small\leftarrow}\{X_\sigma,\pi^\sigma_\rho,\Sigma\}$.

Why the word "thread" is chosen? I mean what's the relation to everyday meaning of thread?

• Probably because they tie together! Maybe it's in anaology with the usage of the term fibre. – Eoin Jul 10 '15 at 5:52
• Just be glad it isn't Kauffman's space terminology. – anakhro Jul 10 '15 at 20:21

I suspect that the metaphor derives from the case in which the directed set $\Sigma$ is linearly ordered. In that case the coordinates $x_\sigma$ are in a sense ‘strung together in a line’ by the maps $\pi_\rho^\sigma$: the maps specify how the thread winds through the spaces $X_\sigma$.