What are some motivating examples of exotic metrizable spaces

Solution 1:

There exist a metric space $X$ containing two open balls $B_1$ and $B_2$ of radii $r_1$ and $r_2$ respectively such that $B_1 \subsetneq B_2$ and $r_1>r_2$.

For example, take $X=[0,+ \infty)$ and $B_1=B(0,1)=[0,1)$ and $B_2=B(1/3,4/5)=[0,1+2/15)$.

Solution 2:

It's well known that any completely metrisable space is a Baire space. From Bourbaki's Topologie générale we have the following example of a metrisable Baire space which is not completely metrisable:

Let $Q := \mathbb Q \times \{0\} \subset \mathbb R^2$ and for each integer $i$ let $K_i := \{(i/n,1/n) \in \mathbb R^2 \mathrel| n \in \mathbb N\} \subset L_{(i,1)}$, where $L_{(x,y)}$ is the ray from $(0,0)$ through $(x,y)$ in $\mathbb R^2$. Then define $X = Q \cup \bigcup_{i\in \mathbb Z} K_i$.

Let $(p/q,0) \in Q$ and define a sequence $(x_n)$ by $x_n := (p/q,1/nq) = (np/nq,1/nq) \in K_{np}$. Then $(x_n)$ will be a sequence in $X\setminus Q$ converging to $(p/q,0)$, so $X \setminus Q$ is dense in $X$.

It's also easily seen that $X \setminus Q$ is discrete and it's open as $Q$ is (sequentially) closed. Thus any dense open set of $X$ must contain $X \setminus Q$, so an arbitrary (and thus in particular a countable) intersection of dense open sets in $X$ is dense.

But $X$ cannot be completely metrisable for if it were, then $Q$—being a closed subspace of $X$—would also be completely metrisable.