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My question concerns the notion of unicoherence, which is a property that a topological space may or may not have. The definition (from Wikipedia) is:

"A topological space $X$ is said to be unicoherent if it is connected and the following property holds: For any closed, connected $A, B \subset X$ with $X = A \cup B$, the intersection $A \cap B$ is connected. For example, any closed interval on the real line is unicoherent, but a circle is not."

Thus it is known that the spaces $\mathbb{R}^n$ with the usual (euclidean) topology are unicoherent, and proofs can be found in various topology texts.

I am interested in knowing whether anyone has any knowledge of methods that might be used to prove that $\mathbb{R}^n$ with non-euclidean topologies are unicoherent (or not). My particular interest is the the classical fine topology from potential theory (the coarsest topology which makes all the subharmonic functions continuous), since if I knew that $\mathbb{R}^n$ were unicoherent, I could possibly make some progress with some work I did some years ago (and conversely, if I knew that it were not unicoherent, then I would at least know that I would be wasting my efforts trying to do so!).

My background is in analysis, and my knowledge of algebraic topology is limited to little more than the contents of any book entitled something like "A first Course in Algebraic Topology", but if anyone could point me in the direction of any results concerning unicoherence in non-euclidean topologies, or any methods that might be appropriate for tackling such questions, I would be grateful.

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See… under "Tumbleweeds" – Will Jagy Nov 8 '12 at 21:51
I am interested in the same question. My approach is to go through a proof that the euclidean space is unicoherent and see how it can be adapted to topologies that are extensions of the euclidean topology. I am at the starting point. In fact, I am trying to find a textbook that has a self-contained proof that euclidean spaces are unicoherent. Can you recommend anything? – user156495 Jun 24 '14 at 9:37

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