Difficult# Defining a Relation with Nets

TOPO-CXKVWB

Let $(X,{\mathcal T})$ be a topological space containing points $x$ and $y$. Define the *subset relation* $\,\leq\,$ on ${\mathcal T}\times {\mathcal T}$ by writing $A\leq B$ if and only if $B$ is a subset of $A$ (and possibly equal to $A$). For any point $z\in X$, denote by $N(z)$ the family of all the open sets that contain $z$. Suppose in addition that there exists $O, O'\in N(x)\setminus N(y)$ and $V, V' \in N(y)\setminus N(x)$ such that $O\cap V\neq \emptyset$ and $O'\cap V' =\emptyset$.

Assume also that $n:{\mathcal T} \to X$ is a function that maps a non-empty open set $U$ to a point $n(U)\in U$, and let $n(\emptyset)=x$ where $x$ is the first point given above.

Find the incorrect statements below. Select **ALL** that apply.