As you can see, $B_1$, $B_2$, $B_3$ are all disconnected and non-overlapping. Let's say they are sets. Can we take the union of $B_1$ to $B_3$ and create a new set called B?
I don't know how to describe these other than things that are fake pseudo topologically disconnected spaces.
Well, I don't know what a "fake pseudo topologically disconnected space" is, but it doesn't matter. Any pair of sets can be unioned together to form a new set, in any circumstance, no matter what. Whether they are disconnected, overlapping, whatever, they can always be unioned together. To "union" two sets just means to form a set consisting of the things that are in each - for example, the union of $\{1, 2, 3\}$ and $\{1, 4, 5\}$ is $\{1, 2, 3, 4, 5\}$. Topological properties have absolutely no effect whatsoever.
Whether or not $B$ is also a "fake pseudo topologically disconnected space" is less clear - sometimes the union of two sets with a certain property doesn't also have that property. And since I don't know what such a space is (I know "topologically disconnected" and "pseudo-topological space", but not "fake") I can't confirm whether or not this property is preserved under unions.