Let $X$ a complex curve and $x\in X$ a point.

We consider the space of divisors $D$ with fixed degree $d$, with no multiplicity outside the point $x$ and with multiplicity less or equal than 2 at $x$?

Why this space is not open?

Now, we consider the same space of divisors $D$ with fixed degree $d$ such that:

$\sum\limits_{x untransversal}m_{x}(D)\leq N.$

where the untransversal points means that $m_{x}(D)\geq 2$.

Is this space open?

Let $X$ a complex curve and $x\in X$ a point.

We consider the space of effective divisors $D$ with fixed degree $d$, whic we know is isomorphic to $X^{d}/S_{d}$ where $S_{d}$ is the symmetric group.

Now, we consider the subspace of divisors $D$ with fixed degree $d$ such that:

$\sum\limits_{x untransversal}m_{x}(D)\leq N.$

where the untransversal points means that $m_{x}(D)\geq 2$.

Is this space open in $X^{d}/S_{d}$?