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.

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