In the plane $X=\mathbb{R}^2$ we consider the family $\tau$ of all the subsets $U$ such that for each point $(a,b)\in U$ it exists $\varepsilon >0$ with $$((a-\varepsilon,a+\varepsilon)\times \{b\}) \cup (\{a\} \times (b-\varepsilon,b+\varepsilon)) \subset U.$$ Study if $\tau$ is a topology in $X$.
Evidently, $X$ and $\emptyset$ are contained in $\tau$.
If we choose $U_1$ and $U_2$ two open sets of $\tau$, the intersection can be the empty set or not. In the first case, $U_1 \cap U_2 \in \tau$. In the second case, the intersection cannot be only a point or a line because of the form of the subsets $U$ described previously. Then, for each point $(a,b)\in U_1 \cap U_2$ it exists $\varepsilon >0$ with $$((a-\varepsilon,a+\varepsilon)\times \{b\}) \cup (\{a\} \times (b-\varepsilon,b+\varepsilon)) \subset U_1 \cap U_2.$$
For an arbitrary collection of open sets $\{U_\alpha\}_{\alpha \in I}$ the idea is similar. If they are disjoint, no problem. And if not, they cut in a subset that cannot be only a point or a line.
I think that it is the idea, but I don't know how to formalize it. Can anyone help me? Thank you.