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Topology

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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.

A

The relation $\,\leq\,$ is both reflexive and transitive.

B

The relation $\,\leq\,$ is symmetric.

C

${\mathcal T} \setminus \{\emptyset\}$ is a directed set with respect to $\,\leq\,$, and the restriction of $n$ to ${\mathcal T} \setminus \{\emptyset\}$ is a net in $X$.

D

$N(x)\setminus \{\emptyset\}$ is a directed set with respect to $\,\leq\,$, and the restriction of $n$ to $N(x)\setminus \{\emptyset\}$ is a net in $X$.

E

$X$ is necessarily countable.

F

${\mathcal T}$ is necessarily uncountable.

G

$(N(x) \cup N(y)) \setminus \{\emptyset\}$ is a directed set with respect to $\,\leq\,$, and the restriction of $n$ to $N(x) \cup N(y)\setminus \{\emptyset\}$ is a net in $X$.

H

$X$ is necessarily uncountable.