![](/files/happy5.png)
Generalized path-connectedness in proximity spaces
Let be a proximity.
A set is connected regarding
iff
.
Conjecture The following statements are equivalent for every endofuncoid
and a set
:
![$ \mu $](/files/tex/12e00f6f7e80e7b1fd1b89a31a7e0abe4c1b1302.png)
![$ U $](/files/tex/3fc3219c567e1da4bea338616076fc2437c024d5.png)
- \item
![$ U $](/files/tex/3fc3219c567e1da4bea338616076fc2437c024d5.png)
![$ \mu $](/files/tex/12e00f6f7e80e7b1fd1b89a31a7e0abe4c1b1302.png)
![$ a, b \in U $](/files/tex/b9afee507ee1fdd9052fada0ecf84737e5f57da8.png)
![$ P \subseteq U $](/files/tex/6bd29b775b103849d9f752b5b7bcb156d949341f.png)
![$ \min P = a $](/files/tex/823bbdacb0b048c6f8f9e75c63e39794dd918e62.png)
![$ \max P = b $](/files/tex/3def450ce9e9804c9d4a8be3ece00739a3893a85.png)
![$ \{ X, Y \} $](/files/tex/69a8a2715ddfa2480ba4a8321219c905c5402338.png)
![$ P $](/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png)
![$ X $](/files/tex/302cdeba125e821f3406302c9789229d48f42ea7.png)
![$ Y $](/files/tex/6e8160788c99301d68bd6cf12fcc0ed07fd138d7.png)
![$ \forall x \in X, y \in Y : x < y $](/files/tex/4b87461ca127a265ebe9b2b7dcf0ccbed211c81e.png)
![$ X \mathrel{[ \mu]^{\ast}} Y $](/files/tex/0ef560be389646efd1fdde5ebc9afc9ac98ee64e.png)
Bibliography
*Question at math.StackExchange.com by Victor Porton
* indicates original appearance(s) of problem.
A proposed lemma
http://math.stackexchange.com/questions/691643/a-lemma-to-solve-a-conjec...
--
Victor Porton - http://www.mathematics21.org