History
In 1920, Bronisław Knaster and Kazimierz Kuratowski asked whether a nondegenerate homogeneous continuum in the Euclidean plane R2 must be a Jordan curve. In 1921, Stefan Mazurkiewicz asked whether a nondegenerate continuum in R2 that is homeomorphic to each of its nondegenerate subcontinua must be an arc. In 1922, Knaster descovered the first example of a homogeneous hereditarily indecomposable continuum K, later named the pseudo-arc, giving a negative answer to the Mazurkiewicz question. In 1948, R.H. Bing proved that Knaster's continuum is homogeneous, i.e., for any two of its points there is a homeomorphism taking one to the other. Yet also in 1948, Edwin Moise showed that Knaster's continuum is homeomorphic to each of its non-degenerate subcontinua. Due to its resemblance to the fundamental property of the arc, namely, being homeomorphic to all its nondegenerate subcontinua, Moise called his example M a pseudo-arc. Bing's construction is a modification of Moise's construction of M, which he had first heard described in a lecture. In 1951, Bing proved that all hereditarily indecomposable arc-like continua are homeomorphic — this implies that Knaster's K, Moise's M, and Bing's B are all homeomorphic. Bing also proved that the pseudo-arc is typical among the continua in a Euclidean space of dimension at least 2 or an infinite-dimensional separable Hilbert space.
Read more about this topic: Pseudo-arc
Famous quotes containing the word history:
“Considered in its entirety, psychoanalysis won’t do. It’s an end product, moreover, like a dinosaur or a zeppelin; no better theory can ever be erected on its ruins, which will remain for ever one of the saddest and strangest of all landmarks in the history of twentieth-century thought.”
—Peter B. Medawar (1915–1987)
“History is the present. That’s why every generation writes it anew. But what most people think of as history is its end product, myth.”
—E.L. (Edgar Lawrence)
“This is the greatest week in the history of the world since the Creation, because as a result of what happened in this week, the world is bigger, infinitely.”
—Richard M. Nixon (1913–1995)