From TopCom, Volume 8, #1
In many branches of mathematics and in applications, one has to deal with families of sets such as closed sets, compact sets, closed (compact) convex sets of a TVS, etc. So there is a need to assign a topology to the space of subsets, called a hyperspace, of a given topological space (X,t). One of the early attempts, in a very general setting, was made by Vietoris [9] in 1922. (See his obituary in [7].) Let CL(X) denote the family of all nonempty closed subsets of the space X.
Let us first see how we can modify the Vietoris topology. We note that if X is metrizable with a metric d, then in the lower Vietoris topology, we can replace the family {Uk} of hit sets by a pairwise disjoint family of open balls {S(xk, e)} where xk in Uk and e > 0. Moreover, by using Zorn's Lemma, we can get, for each A in CL(X) and e > 0, a maximal discrete family Q = Q(A, e) subset A with the properties
Next we change the upper Vietoris topology by requiring the members E of CL(X) to be far from a closed set C, i.e., d(E,C) > 0. We call C the far set. In this way we get a new topology:
Clearly E is far from C implies E misses C but the converse is not true. For example, the set E = { (x,y) in R2 : y = x-1, x > 0, y > 0} misses the set C = { (x,y) in R2 : y => 0} but E is not far from C.
Combining (4) and (5) we get, for a metric space (X,d), a modified hit-and-miss topology viz.:
We are familiar with the Hausdorff metric dH on CL(X), which is a standard exercise in most text books. This was discovered by Hausdorff about ten years before the discovery of Vietoris topology and an equivalent metric was studied by Pompeiu in 1905. For closed sets A, B in CL(X), the Hausdorff metric is defined by:
(*) dH(A,B) = inf {e > 0 : A subset S(B,e), B subset S(A, e)}or dH(A,B) = infinity if no such e exists.
On the face of it, the Hausdorff metric topology t(dH) looks to be of quite a different genre than the Vietoris topology t(V). However, it was shown recently that t(dH) is similar to t(V) and, in fact, it equals the discrete hit-and-far topology (6).
Theorem [4]. t(dH) = t(Ld) v t(d+).
With the above representation of the Hausdorff metric topology t(dH), it is easy to compare it with the Vietoris topology t(V). From (4) and (5) it follows that they are equal if and only if X is compact.
What is common to C*-algebras and mines? It is the Fell topology! The Vietoris topology was modified by Fell ([3]) to solve a problem in C*-algebras. In the upper Fell topology t(F+), instead of taking all closed sets as miss sets, as in the upper Vietoris topology, Fell took only compact sets as miss sets. He left the lower Vietoris topology unchanged.
It is a great surprise that G. Matheron, recently rediscovered the Fell topology while working on problems of mines! ([8])
The formula (*), for the Hausdorff metric, can also be expressed as
dH(A,B) = sup {|d(x,A) - d(x,B)| : x in X}.From the above expression, it is clear that a sequence {Ak} of closed sets converges to a closed set A in the Hausdorff metric topology if and only if the sequence {d(x,Ak)} converges uniformly on X to d(x,A). For tackling problems in Convex Analysis, Wijsman ([10]) replaced uniform convergence by pointwise convergence.
The resulting Wijsman topology can be expressed as
Obviously, "A misses the bigger ball S(x, a)" implies "A is far from the smaller ball S(x,e)". So the upper Wijsman topology t(W+) is a far-miss topology. A typical neighborhood of A in CL(X) in t(W+) consists of sets E in CL(X), which miss a ball B that is far from A. Here the far-miss sets can be open or closed balls, since two sets are far iff their closures are far. Thus the Wijsman topology (8) is a hit-and-far-miss topology.
There are many other hypertopologies. They have lower parts similar to the lower Vietoris topology, sometimes containing discrete or locally finite families of open sets as hit sets. The upper parts are similar to the upper Fell topology, and the miss sets can be balls in a metric space or, in general, any family D subset CL(X) ([6]).
There is a vast ever growing literature in hyperspaces and there are numerous applications to Mathematical Economics, Optimization, Convex Analysis, Functional Analysis, Mines, etc. For unification of all hypertopologies see [2] and for a brief history see [5]. We give a short list of references below and the interested readers can get further information from them.
Date: May 7, 2003.