The geometry of convergence in numerical analysis

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