Exercise 2.1.6 (Metrizable spaces are separable)

Question

\begin{enumerate}[label=(\alph*)]
\item Show that if $X$ is a metrizable topological space and if $p$ and $q$ are distinct points of $X$, then there are open sets $U$ and $V$ such that $p \in U$, $q \in V$, and $U \cap V = \emptyset$.
\item Let X be an infinite set and let $\mathcal{T}$ be the cofinite topology on $X$ (\href{./exercise_2-1-2}{Exercise 2}). Prove that the property described in part (a) does not hold for the open sets in $X$ and hence that $X$ with the cofinite topology is not metrizable.
\end{enumerate}

Elu Thingol · Accepted Answer

\begin{enumerate}[label=(\alph*)]
\item Let $d$ be the corresponding metric on $X$. Then the open balls 
$$U:=B\left(p, \dfrac{d(p,q)}{3}\right), \quad V:=B\left(q, \dfrac{d(p,q)}{3}\right)$$
centred at $p$ and $q$ of radius $d(p,q)/3$ are obviously disjoint by triangle inequality.

\item If $X$ is infinite and if $U$ and $V$ are nonempty open sets in the cofinite topology (i.e., such that $X\setminus U$ and $X \setminus V$ are finite), then the complement $X\setminus (U \cap V) = (X \setminus U) \cup (X \setminus V)$ of $U \cap V$ must be finite as well, meaning $U \cap V$ is infinite and therefore not empty.
\end{enumerate}

Exercise 2.1.6 (Metrizable spaces are separable)

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Exercise 2.1.6 (Metrizable spaces are separable)

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