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An interpretation of betweenness on a set is gap free if each two distinct points of the set have a third point between them. In this paper we are interested in gap free betweenness relations naturally induced by the topology of Hausdorff continua. In particular, we say c lies between a and b in the K-interpretation precisely when every subcontinuum that contains both a and b also contains c. We explore the connection between K-gap freeness and hereditary unicoherence.
Bankston, Paul, "When Hausdorff Continua Have No Gaps" (2014). Mathematics, Statistics and Computer Science Faculty Research and Publications. 188.