TITLE: Do you have examples of such "transitive" elements?
QUESTION [8 upvotes]: (I've asked the same question at the MSE, so far with no answers, so I thought I'd try it here as well. If there's some clash with any site rules, please let me know and I'll abide.)
Let $A$ be a set equipped with a binary reflexive and symmetric relation $\smile$ (such relations have been called tolerances and similarities in the past). I'm interested in meaningful mathematical instances of the following property for an element $a \in A$:
$$
\forall_{b,b' \in A} (b \smile a \land a \smile b' \rightarrow b \smile b') \quad (\star)\ .
$$
We may think of $a$ as a "transitive" element, a "transition go-between" for the otherwise not necessarily transitive relation $\smile$. Depending on the choice of carrier and relation, there may be many such elements, just one, or maybe none.
My questions: Is the property $(\star)$ known and studied? Is there a name for it? Are there natural examples of it?
I've personally thought of a couple (I've even elaborated on a simple one in MSE) but I'd rather not influence any possible answers here --- let alone, I'm not sure I'm satisfied with what I've found.

REPLY [3 votes]: Interesting feedback, thanks.
It seems to me that the property I'm talking about is indeed meaningful in various settings, but apparently doesn't come up so often, or rather, in an interesting or useful enough fashion, to warrant special attention.
In any case, here are some further elementary examples that I'm aware of.

The one I've already written in my MSE question: for the standard tolerance relation on the euclidean plane, namely $x \smile y$ if and only if their distance is bounded by some fixed constant, there are no such points. But if we restrict the plane to some cone $A$ which is narrow enough (angle at most $60^o$), then the apex of the cone is "transitive".
Again on the euclidean plane, take $A$ to be the complement of a disc, boundary included (a non-convex set) and say that $x \smile y$ if and only if they are joined by a line segment entirely within $A$. The "transitive" elements here are the points of the circle (everything related to them has to be above the corresponding tangent).
Take $A$ to be the set of all sequences in the interval $[0,1]$; say that $(x_n) \smile (y_n)$ when $(x_n)$ and $(y_n)$ share an accumulation point. Then the converging sequences are the "transitive" ones.