Please consider:
$$A = (-∞, x)\,\,\,; \,\,\,B = (x,∞)$$
How to prove that these intervals are adjacent, that is, that point $x$ does not constitute a distance between these sets?
In other words, if there are three points $i$, $j$, $k$ that immediately follow each other, and $j$ is between $i$ and $k$, how do I prove that in the case that $j$ is missing (i.e. not part of the interval), there isn't a "gap" between i and $k$?