Let $(\Omega,\Sigma,\mathbb{P})$ be a complete probability space, $B\subseteq X$ be a non-empty Borel subset of a polish space $X$, $A$ be an uncountable indexing set, and $\{X\_{\alpha,n}\}\_{a \in A, n \in \mathbb{N}}$ a set of $X$-valued random elements with the following properties:


* For every pair of distinct $\alpha,\alpha'\in A$ and every $\omega \in \Omega$ there does not exist $n,n'\in\mathbb{N}$ satisfying 
$$
X\_{\alpha,n}(\omega) = X\_{\alpha',n'}(\omega).
$$
* For every $\alpha \in A$, and every non-empty open subset $O'\subseteq X$, $0<\mathbb{P}\left(\bigcup\_{n \in \mathbb{N}}\{X\_{\alpha,n} \in O'\}\right)$.


Then can we conclude that $1=\mathbb{P}\left(\bigcup\_{n \in \mathbb{N},\alpha \in A}\{X\_{\alpha,n} \in B\}\right)$?



Getting almost certainty from uncountably many low-probability events
Here is a "very regular" counterexample:


Let $X=\mathbb R$, $A:=(0,1)=:B$, and $X\_{a,n}:=Z+a$ for all $a\in A$ and $n\in \mathbb R$, where $Z\sim N(0,1)$. Then all your conditions hold. However, 
$$P\Big(\bigcup\_{n\in\mathbb N,a\in A}\{X\_{a,n}\in B\}\Big)=P(|Z|<1)\ne1.$$


