How do I prove this given set is open in the space of probability measures?

Question

Let $X$ be a separable metric space and let $Prob(X)$ be the set of Borel probability measures on $X$, equipped with the weak* topology.

Here are the facts that I know:

1. The collection $\{\lambda\in Prob(X): \lambda(F) < \mu(F)+\epsilon\}$'s form a subbasis for the topology on $Prob(X)$, where $F$'s are closed in $X$ and $\mu\in Prob(X)$.

2. The collection $\{\lambda\in Prob(X): \lambda(G)> \mu(G) -\epsilon\}$'s form a subbasis for the topology on $Prob(X)$, where $G$'s are open in $X$ and $\mu\in Prob(X)$.

3. The collection $\{\lambda\in Prob(X): |\int_X f d\lambda - \int_X fd\mu|<\epsilon\}$'s form a subbasis for the topology on $Prob(X)$ where $f\in C_b(X)$ and $\mu\in Prob(X)$.

4. The topology on $Prob(X)$ is metrizable and second-countable.

My question is the following:

Let $\mu\in Prob(X)$ and $A$ be a Borel subset of $X$ such that $\mu(cl(A)\setminus Int(A))=0$ (which is called the $\mu$-continuity set).

In this case, how do I prove that the set $\{\lambda\in Prob(X): |\lambda(A)- \mu(A)|<\epsilon\}$ is open in $Prob(X)$?

Define the evaluation map $\pi_A(\lambda):=\lambda(A)$. By the Portmanteau teorem, we know that $\pi_A$ is continuous at $\mu$. However, this does not gurantee that the inverse map $\pi_A^{-1}((\mu(A)-\epsilon,\mu(A)+\epsilon))$ is an open set, because it is just continuous at a point $\mu$.. I am stuck at showing this. How do I prove this? Thank you in advance!