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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the details in this proof scheme, I will not do it myself because internally in my brain I solved it using some yet unpublished concepts.)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.)

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It's a standard result that if $X$ is a topological space and $(A_i)$ is a family of connected subsets of $X$ with $\bigcap A_i\ne\emptyset$ then $\bigcup A_i$ is connected. Indeed this is often set as an exercise. (Hint: think of continuous functions from the union to a two-element discrete space.) – Robin Chapman Jul 13 '10 at 19:05
In the usual usage, every uniform space is a topological space, so something that holds for all topological spaces should hold for all uniform spaces. – Charles Staats Jul 13 '10 at 20:09
@porton: but the word "uniformly" does not appear in your conjecture. Was that your intent? – Pete L. Clark Jul 13 '10 at 22:26
Why would anyone give a question with at least 8 downvotes a bounty? – Joseph Van Name Jun 27 '13 at 0:16
The only possible reason to give this sort of question a bounty is so that this question can get more attention and hence more downvotes. – Joseph Van Name Jun 27 '13 at 0:25

This result follows immediately from the main result in [S. G. Mrówka and W. J. Pervin, On Uniform Connectedness, Proceedings of the American Mathematical Society, Vol. 15, No. 3 (Jun., 1964), pp. 446-449], namely, that a uniform space is uniformly connected in your sense iff every uniformly continuous function from it to a discrete space is constant.

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