# Does convergence in probability w.r.t. a topology make sense?

Let $(S,d)$ be a separable metric space. A sequence of random variables $X_n$ is said to converge in probability to $a \in S$ if and only if for all $\varepsilon > 0$ $$P(d(X_n,a) > \varepsilon) \to 0,$$ as $n \to \infty$.

Would this make sense in a topological space? Let $S$ be a topological space instead and, instead of the condition above. require that for any open $U \ni a$ $$P(X_n \in U) \to 1.$$

Does this type of concept exist? Would it make any sense? What I have in mind is a probabilistic version of convergence of operators on Hilbert space w.r.t. the weak operator topology.

Your proposed definition of convergence in probability that $\text{plim}_{n\to\infty}X_n=x$ if $P(X_n\in U)\rightarrow 1$ for all open neighbourhoods, $U$ of $x$, makes sense, and can be found in the literature. Two examples I've seen recently are Liese and Vajda (1995), or in Heijmans and Magnus (1986). In both cases the definition is given in the context where the random variable is assumed to take values in a metric space, but that assumption is not needed for the definition to make sense.