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The geometry of convergence in numerical analysis

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  • The domains of mesh functions are strict subsets of the underlying space of continuous independent variables. Spaces of partial maps between topological spaces admit topologies which do not depend on any metric. Such topologies geometrically generalize the usual numerical analysis definitions of convergence.

    Mathematics Subject Classification: Primary: 65LXX; Secondary: 54B20.

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  • Figure 1.  Illustrating convergence in the lower and upper Vietoris topologies. Right: a subbasic neighbourhood of a subset $ A $, in the upper Vietoris topology, is defined by an open set $ U $. $ A $ is contained in $ U $ and the green sets are in the subbasic neighbourhood are contained in $ U $. As $ U $ shrinks, every point in the green sets in drawn to some point in $ A $—everything approximable is in $ {\rm cl}A $. At left, in the lower Vietoris topology, the green sets only have to intersect $ U $, and shrinking $ U $ around a single point of $ A $ generates approximations when the green sets also meet $ U $—everything in $ A $ is approximable

    Figure 2.  Left: a neighbourhood of the geometric topology is defined by an open set $ U $, a compact set $ K $, and open sets $ V_i $. Containment within $ U $ has to occur only inside the compact set $ K $, with effect that a convergent sequence of subsets $ \langle A_i\rangle $ has $ K\cap A_i $ finally contained in $ U $, say for some $ i\ge N $, but larger $ K $ require larger $ N $. As shown the set $ A $ is inside the neighbourhood because its intersection with $ K $ is contained in $ U $ and contacts each $ V_i $. Smaller $ U $, and larger $ K $, and more and smaller $ V_i $, correspond to a smaller more restrictive neighbourhood. Right: a basic neighbourhood of a compact set $ B $ in the geometric topology. A subset of $ X $ is inside such a neighbourhood if it is contained in $ U $ and contacts each $ V_i $

    Figure 3.  Left: the discrete approximations $ y_i $, $ i = 1, 2, 3\ldots $ (circles) are limiting to a continuous $ y $. Shown (squares on red curves) are three selections of subsequences from the graphs of $ y_i $. Every such subsequence converges to the graph of $ y $, and that graph is the limit of such subsequences. Right: an open neighbourhood of the red graph is defined by a compact set $ K $, an open set $ U $, and open sets $ V_i $. $ K $, which may be restricted to a product of compact sets, may be thought of as a frame within which proximity to the graph is controlled by $ U $. The other black curves are in the neighbourhood because they also contact the $ V_i $. Larger $ K $, smaller $ U $, and more and smaller $ V_i $, correspond to smaller neighbourhoods

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