# Set System with Topology; TOP-set system

By | September 18, 2020

Here we mention the definition of Set System that came in the paper “Epsilon-approximations and epsilon-nets” by Nabil H. Mustafa and Kasturi R. Varadarajan;

A pair $\sum = (X, \mathcal{R})$, where $X$ is a set of base elements and $\mathcal{R}$ is a collection of subsets of $X$ , is called a set system.

Question: ‎If ‎$(X, ‎\mathcal{R}‎) ‎$ ‎be a set system‎, can ‎we ‎assign‎ a‎‎ topology‎ ‎$‎\tau_{X}^{‎\mathcal{R}‎}‎ ‎$ ‎to ‎the ‎subspace‎

$‎‎\widetilde{\mathcal{R}} = ‎\bigg \{ \bigcup R \ | \ R \in \mathcal{R} ‎‎‎‎\bigg \} ‎$ of ‎$X ‎$ ‎, that ‎ ‎$‎\mathcal{R} \subseteq \tau_{X}^{\mathcal{R}}‎ ‎$ ‎‏, ‎and no other topology ‎ ‎$‎\tau‎^{'} ‎$ on‎ ‎$\widetilde{\mathcal{R}} ‎$ { ‎‎‎$‎\mathcal{R} \subset ‎\tau^{'} ‎$‎ ) ‎exists that ‎$\tau^{'} \subset \tau_{X}^{\mathcal{R}} ‎$ ‎‎?

If ‎so, ‎is‎ ‎$\tau_{X}^{\mathcal{R}}$ ‎unique‎?

The answer is likely Yes; by Constructing Topologies, consider

$\Gamma (\mathcal{R}) = \bigg \{ \tau: \tau \ \ topology \ on \ X \ with \ \mathcal{R} \subset \tau \bigg \}$

Now ‎the ‎intersection

$TOP(\mathcal{R}) = \bigcap_{\tau \in \Gamma} \tau$

‎is th‎e ‎smallest ‎(weakest) ‎topology ‎among ‎all ‎topologies ‎with ‎respect ‎to ‎which ‎all ‎sets ‎in‎ ‎$\mathcal{R}‎$ ‎are ‎open.‎

Then we call $(X , \mathcal{R} , TOP(\mathcal{R}) )$ a set system with a topology and show it by TOP-set system.

How ‎about ‎assigning a $\sigma$-algebra‎‎‎ ‎$\Omega_{X}^{\mathcal{R}}$ ‎to‎ $\widetilde{\mathcal{R}}$ with alike properties mentioned above?

Probably Yes.