Fix an open set $D \subset \mathbb{R}$ and consider the following set of probability measures, where $\mathfrak{M}(\mathbb{R})$ is the space of probability measures on $\mathbb{R}$ endowed with the weak topology:
$$ B = \{ P \in \mathfrak{M}(\mathbb{R}) \: : \: \text{supp}(P) \subset D \} $$
Is $B$ open in the weak topology?
The support of a probability measure on a separable metric space (like $\mathbb{R}$) is the smallest closed subset of probability 1.
Try this. Let $D = (-1,1)$, an open set. Let $\mu_n$ be the probatility measure with mass $1/n$ at point $5$ and mass $(n-1)/n$ at point $0$. Then $\mu_n$ converges to the unit-point-mass at $0$. Although all $\mu_n$ are outside $B$, the weak limit is in $B$.