Serre duality: Difference between revisions

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In mathematics, a '''unicoherent space''' is a [[topological space]] <math>X</math> that is [[connected space | connected]] and in which the following property holds:
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For any closed, connected <math>A, B \subset X</math> with <math>X=A \cup B</math>, the intersection <math>A \cap B</math> is connected.
 
For example, any closed interval on the real line is unicoherent, but a circle is not.
 
If a unicoherent space is more strongly hereditarily unicoherent (meaning that every subcontinuum is unicoherent) and [[Connected space#Path connectedness|arcwise connected]], then it is called a [[Dendroid (topology)|dendroid]]. If in addition it is [[Locally connected space|locally connected]] then it is called a [[Dendrite (mathematics)|dendrite]]. The [[Phragmen–Brouwer theorem]] states that, for locally connected spaces, unicoherence is equivalent to a separation property of the closed sets of the space.
 
==References==
*{{MathWorld|urlname=UnicoherentSpace|title=Unicoherent Space|author=Insall, Matt}}
 
[[Category:General topology]]
[[Category:Trees (topology)]]
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Latest revision as of 22:44, 25 September 2014

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