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Let δ is a proximity.

I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y.

Question: Let A and B are sets with non-empty intersection. Let both A and B are connected. Prove or give a counter-example that A∪B is also connected.

(This question arouse as a special example of a more general theorem. I spend may be half of hour attempting to prove it and after these my efforts failed, I desire to share this question.)

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Consider $X\cap A$ and $Y\cap A$, starting from a partition $\lbrace X,Y\rbrace$ of $A\cup B$. If both intersections are nonempty we are done, as $(X\cap A)\delta(Y\cap A)$. Otherwise, $A\subseteq X$, say, but then $X\cap B$ and $Y\cap B$ are nonempty and we find $(X\cap B)\delta(Y\cap B)$. In either case $X\delta Y$.

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  • $\begingroup$ Thanks for your solution. In fact I solved myself, but my solution was less beautiful. KP Hart, do you want that I would refer to you in the article (about connectedness) which I am writing? $\endgroup$
    – porton
    Commented Jun 14, 2010 at 22:13
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    $\begingroup$ There's no need to credit anyone I think as it follows almost straight from the definitions. Maybe check this book (ams.org/mathscinet-getitem?mr=278261) to see if it isn't simply well known. (By the way, it suffices that $A\delta B$: either one of $A$ and $B$ meets both $X$ and $Y$ and we're done or $\lbrace A,B\rbrace=\lbrace X,Y\rbrace$, in which case we have $X\delta Y$ too.) $\endgroup$
    – KP Hart
    Commented Jun 15, 2010 at 13:26

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