I will call a set uniformly connected regarding some uniform space when it is connected regarding every entourage of this uniform space (entourages are considered as digraphs and it is taken strong connectedness regarding these digraphs).
Is the following true? (I need a proof or a counter-example.)
Conjecture If S is a collection of uniformly connected sets and ∩S≠∅ then ∪S is uniformly connected.
Note that uniform space may be replaced with an arbitrary filter on the set of binary relations on some set (I call filters on binary relations reloids, reloids are a generalization of uniform spaces). You may read my theory of reloids (including some results about connectedness regarding reloids) in this my draft article at my site.
Oh, you may close this question. The solution I found myself is too easy:
We can prove that ∪S is connected for every entourage (considered as a digraph). So it is connected regarding the uniform space. (Somebody may fill the details in this proof scheme, I will not do it myself because internally in my brain I solved it using some yet unpublished concepts.)
(I though about solution in more general terms and so missed this simple solution. Sorry me for pollution of MO with too simple questions.)