= Solution
With the quotient realization defined in part (i), \b[the answers are no and no]. Write $W=|K|$ as countably many closed arms $[0,1]_n$ with all their zero endpoints identified to $v$.
First, the outer vertices $D=\{u_n:n\geq1\}$ form a <closed discrete subset>. Its intersection with every closed arm is the closed singleton at coordinate one, so $D$ is closed by the weak topology. The terminal half-arm $(1/2,1]_n$ is an open neighbourhood of $u_n$ meeting $D$ only there. Thus $D$ is infinite, closed and discrete. Since <closed discrete subsets of compact spaces are finite>, $W$ is not compact. A <homeomorphism> to a compact subset of the plane would make $W$ compact, which is impossible.
For the stronger obstruction, suppose $(U_n)$ were a countable <neighbourhood basis> at $v$. Each $U_n$ contains an initial interval in arm $n$, so choose $x_n\in U_n$ on that arm with coordinate $t_n\in(0,1)$. Define $V$ to meet arm $n$ in $[0,t_n/2)$. It is an open neighbourhood of $v$, because each of these intersections is open in the closed interval $[0,1]$. But $x_n\in U_n\setminus V$ for every $n$, so no $U_n$ is contained in $V$. This contradicts the basis condition.
Hence $W$ is not <first countable>. Every metric space is first-countable, with local basis of balls of radius $1/n$, and this property passes to subsets with the <subspace topology>. Thus $W$ is not homeomorphic to any subset of $\mathbb R^2$. This is the <weak realization of a countable star>.
The realization convention matters. For the <metric realization of a simplicial complex> using Euclidean distance on barycentric coordinates, the second answer would be \b[yes]: map the point on arm $n$ at parameter $t$ to $te^{i/n}$. Each ray is isolated away from zero, and at zero both spaces have neighbourhood bases given by a single radius on all arms. This gives the <metric realization of a countable star> as a planar subset, still noncompact. It does not give the quotient topology used in part (i).
\Image[/past-exam-of-the-topology-course-of-the-university-of-oxford/2017-countable-star-topologies.png]
{title=Independent arm radii in a weak star and uniform radii in a metric planar fan}
{description=Only finitely many arms are drawn. The left panel is a schematic of a weak neighbourhood and the diagonal points excluded from it, rather than an embedding of the whole weak realization. The right panel is a finite part of the planar metric realization; its limiting positive real ray is omitted.}
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