# $T_1$-spaces vs $T_1$-hypergraphs

Let us say that a hypergraph $$H=(V,E)$$ is $$T_1$$ if for $$x\neq y$$ there is $$e\in E$$ such that $$e\cap\{x,y\} = \{x\}$$.

Note that for any $$T_1$$-space $$(X,\tau)$$ the topology $$\tau$$ contains the cofinite topology (the collection of sets with finite complement, along with $$\emptyset$$).

With $$T_1$$-hypergraphs, the situation does not seem as pleasant as this.

Question. If $$H=(V,E)$$ is a $$T_1$$-hypergraph, is there $$E_1\subseteq E$$ such that $$(V, E_1)$$ is $$T_1$$, but whenever $$E_2$$ is a proper subset of $$E_1$$, then $$(V,E_2)$$ is no longer $$T_1$$?

• Thanks for putting this effort in! Mar 1, 2019 at 13:19

Counterexample:

For $$a\in\mathbb Q$$ let $$L_a=\{x\in\mathbb Q:x\lt a\}$$, $$R_a=\{x\in\mathbb Q:x\gt a\}$$.

Let $$H=(\mathbb Q,E)$$ where $$E=\{L_a:a\in\mathbb Q\}\cup\{R_a:a\in\mathbb Q\}$$.

Clearly $$H$$ is a $$T_1$$-hypergraph. Consider any $$E_1\subseteq E$$ such that $$(\mathbb Q,E_1)$$ is $$T_1$$. Choose $$a\in\mathbb Q$$ so that $$L_a\in E_1$$. Let $$E_2=E_1\setminus\{L_a\}$$, a proper subset of $$E_1$$. I claim that $$E_2$$ is $$T_1$$.

Assume for a contradiction that $$E_2$$ is not $$T_1$$. Then there exist $$x,y\in\mathbb Q$$ such that $$x\ne y$$ and there is no $$e\in E_2$$ such that $$e\cap\{x,y\}=\{x\}$$. Since $$E_1$$ is $$T_1$$, we must have $$L_a\cap\{x,y\}=\{x\}$$, i.e., $$x\lt a\le y$$. Choose $$z\in\mathbb Q$$ so that $$x\lt z\lt a$$. Since $$E_1$$ is $$T_1$$, there is some $$L_b\in E_1$$ such that $$L_b\cap\{x,z\}=\{x\}$$, i.e., $$x\lt b\le z\lt a\le y$$. But then $$L_b\in E_2$$ and $$L_b\cap\{x,y\}=\{x\}$$, a contradiction.

P.S. Here is an example where the edges of the hypergraph are finite sets: $$H=(V,E)$$ where $$V=\omega=\{0,1,2,3,\dots\}$$ and $$E=\{\{0,1\},\{0,2\},\{0,3\},\{0,4\},\dots\}\cup\{\{1,2\},\{1,2,3\},\{1,2,3,4\},\dots\}.$$

• In the second line, wouldn't some notation like $R_a$ be better for the latter set? Because right now it's unclear if $L_c$ would mean the latter or the former. Feb 26, 2019 at 11:43