A subset S of a Euclidian space M is closed if and only if it is true that if an infinite sequence of points from S has a limit point P in M, then that limit point is also in S.

A subset S of a Euclidian space M is bounded if and only if there exists a circle (however large) such that all of S is inside the circle.

Sets which are both closed and bounded are said to be compact. The closed interval [0,1] considered as a subset of the real line is compact. The open interval (0,1) is not compact because it is not closed. The entire x-axis, considered as a subset of the plane, is not compact because it is not bounded, but the circle of radius 2000 about the origin is compact.

To construct a hyperspace, we first take a compact subset S of a Euclidian space, and consider it as space by itself ... throw away the rest of the points in the Euclidian space. Two compact subsets A and B of S are close to each other provided that every point in each of the subsets is close to some point in the other subset. Thus the two sets A and B in Figure 1 are not close, because the point x in A is far away from every point in B, whereas the two sets A and B in figure 2 are close to each other.

In this situation, the distance between A and B is the smallest number d such that every point of A is within the distance d of some point of B and every point of B is within the distance d of some point of A. This distance is called the Hausdorf distance.

Let S be a compact subset of a Euclidian space. A hyperspace of S is a new space in which the points are compact subsets of S and the distance between points is found using the Hausdorf distance.

To obtain a hyperspace, we can use any collection of compact subsets we wish. There are three which have their own symbols: H(S) denotes the hyperspace of all compact subsets of S; C(S) denotes the hyperspace of all connected compact subsets of S, and Fn(S) denotes the hyperspace of all subsets of S containing exactly n points. H(S) is sometimes denoted 2^S .  Note that a single point may be considered as a subset of S, so that F1(S) is the same space as S.

A geometric realization of a hyperspace is a picture of the hyperspace in which all the observable features are consistent with the properties of the hyperspace.  Obviously, there may be many geometric realizations for a given hyperspace since, in topology, distances are irrelevant;  only the preservation of limit points is considered relevant.  And, of course, there may be no useful geometric realization. 

A very useful geometric realization for hyperspaces (if available) is one in which the points of the hyperspace can be given coordinates on an appropriate coordinate system, and the coordinates can be obtained directly from the hyperspace itself. 

Interesting questions occur when one has a relatively simple space and some limitation on the compact sets which are the points in the hyperspace. As a first example, let S be the unit interval [0,1] from the real line. Then C(S) is relatively easy to construct: every connected compact subset of S is also an interval or a point, so can be identified completely by its endpoints. For example, [0.22, 0.76] is such an interval. For a single point, say 0.2, use the 'interval' [0.2,0.2].  However, this is a pair of numbers, and thus could be considered as a set of coordinates.  The restrictions on the coordinates (a,b) are that a and b are in the interval [0,1] and a is less than or equal to b. 

This allows a nice geometric realization of C(S) (or C(I), since the unit interval [0,1] is commonly denoted I ):  the triangle in the plane with vertices (0,0), (1,0) and (1,1) contains exactly the points whose coordinates are obtained from C(I).  Furthermore (the essential part) one can see that two points in this triangle are close to each other if and only if the corresponding sets in the hyperspace are close.  Thus the triangle is a geometric realization of the hyperspace C(I). 

Given a space M, a topological hyperspace of M is a new space obtained by considering a collection of subsets of M to be points and defining the distance between these new points in this new space in terms of the original distance on the original space. In order to be more specific, it will be necessary to give some intial definitions in terms of compact metric spaces, but for those who are not familiar with the abstract approach, most of what will be done can be understood simply by thinking about the real line, the plane, and three-dimensional space.

A metric space (M,d) is a collection of points M which has a distance function d defined for the points. The distance function must have three properties: given any three points x, y and z in the space,

1) d(x,y) = 0 if and only if x = y

2) d(x,y) = d(y,x)

3) d(x,y) + d(y,z) <=  d(x,z)  (<= means 'less than or equal to')
 

If P is a point in the metric space, and e is a positive number, then the ball about P of radius e, denoted B(p;e) is defined to be the collection of all points closer to P than the distance e. If A is a collection of points of M (i.e. a subset of M), then P is said to be a limit point of A if and only if, given any positive number e (especially a very small one), there is a point Q in A which is not equal to P and which is in B(P;e).

The real line is a familiar example of a metric space; the distance function d(x,y) = |x - y| is the standard distance function. The Cartesian plane -- the plane with the usual coordinate system -- is another example, using the familiar distance formula which gives the distance between P = (x₁,y₁)
and Q = (x₂,y₂) with the formula:

In order to prove that these are actually distance functions, it is necessary to prove that the three properties given above are actually true for these functions. The third property, called the triangle inequality, is surprisingly difficult to prove for a beginner -- both for the line and for the plane. Give it a try and see for yourself.

Three dimensional spaces, and the higher dimensional Euclidian spaces are also metric spaces.

An interval on the line is the collection of all points between two points; the interval is an open interval if it contains neither end point and closed if it contains both end points.

In order to visualize the situation, you should notice that the ball about a point on the line is an open interval (i.e. does not include the end points); in the plane, it is the interior of a circle and in three space it is the interior of a sphere -- the source of the name. Also notice that the two end points of an interval on the line are limit points of the interval. Thus a closed interval contains all of its limit points whereas an open interval does not. Check this out with some drawings.

A subset S of a metric space (M,d) is said to be compact if and only if every infinite subset of S has at least one limit point which is in S. The collection of whole numbers on the real line is an infinite set which has no limit points at all, hence the real line is not compact. The open interval (1,2) consisting of all points between 1 and 2, not including the endpoints, contains the infinite sequence of points 1.1, 1.01, 1,001, 1.0001, ... which has a limit point, namely 1, but that limit point is not in the open interval, so this open interval is not a compact set.

A subset S of a metric space (M,d) is closed if every limit point of the subtset is in the set. Thus the open interval (0,1) is not a closed subset of the real line, but the closed interval [1,2], which includes the end points, is a closed subset of the real line.

A subspace S of a metric space M is a subset of M which is considered as a space by itself, using the same distance function as was used for M. Roughly speaking, it is a space you get by selecting a subset S of M and throwing away all of the points which are not in S. The important point to notice is that the points that are thrown away no longer exist, so they cannot be limit points. Thus the subspace (1,2) of the real line is closed ... it contains all the points which exist, therefore must contain all the limit points. Now, however, the sequence 1.1, 1.01, 1.001, ... does not have a limit point, so this subspace is not compact.

A well known theorem says that a subset S of a Euclidian space (i.e. the line, the plane, three dimensional space, etc.) is compact if and only if S is closed and bounded ... bounded means that there is some distance d such that no two points in S are further apart than d. Furthermore, the subspace S is compact if and only the subset S is compact in the original space.

We are now ready to construct hyperspaces. We start with a compact metric space M and consider the collection H(M) consisting of all the subsets of M which are compact.

The major theoretical question on hyperspaces was to determine the hyperspace of the unit interval. 

"Probably the first result in the direction of 2^I was by L. Vietoris when he proved ...[1922] that if X is a Peano continuum, then so is [the hyperspace of X] ..... In his earlier paper, Wojdyslawski specifically asked if [the hyperspace of the unit interval is homeomorphic to the Hilbert cube]. Professor Kuratowski has informed us that the conjecture .... was well know to Polish topologists in the 1920's."[5]

This was finally done by Schori and West [5] and generalized by Schori and Curtis [7].

The Curtis-Schori Theorem:

If X is a Peano continuum then:
1) 2^X is the Hilbert cube
2)C(X) is the the Hilbert cube if X contains no free arc
3) C(X) × I is homeomorphic to the Hilbert Cube

Though their proof, and their methods were successful, they were quite complex,

"The Curtis-Schori-West techniques of proof involve the delicate use of inverse limits, subtle and complicated maneuvers with refinements of partitions, and what were, at the time, fairly new results about infinite dimensional topology"[2]

An identification theorem for the Hilbert Cube by Toruncczyk [9] made the proof much easier.

Torunczyk's Theorem: [9] (paraphrase)

The compact continuum X is an absolute retract and the identity map on X is a uniform limit of Z-maps

if and only if

X is the Hilbert cube

By using Torunczyk's Theorem, the problem of identifying the hyperspace of the unit interval (or, indeed, the problem of identifying many hyperspaces) is reduced to showing that the hyperspace has the listed properties, and is therefore the Hilbert Cube.

For spaces in the plane and in 3-space, and particular for simple graphs, the hyperspaces C(X) were investigated by R. Duda [10-12] who gave many characterizations and methods.

1. F. Hausdorff, Mengenlehre, Walter de Gruyter & Co., Berlin, 1927

2. Alejandro Illanes and Sam B. Nadler, Jr., Hyperspaces Fundamentals and Recent advances, Marcel Dekker, Inc., New York, (1999)

3. M. Wojdyslawski, "Sur la contractilite des hyperespaces de continus localement connexes", Fund. Math., 30 (1938) 247-252

4. K. Kuratowski, Topology, Vol 2, Acad. Press, New York 1968 {attributes to Sierpinski 1937}

5. A. Pelczynski, "A remark on spaces 2^X for for zero-dimensional X", Bull. Pol. Acad. Sci. 13 (1965) 85-90

6. R. M. Schori and J. E. West , "2^I is homeomorphic to the Hilbert Cube", Bull. Amer. Math. Soc. 78, (1972) 402-406

7. R. M. Schori and J. E. West, "The hyperspace of the closed unit interval is a Hilbert cube", Trans. Amer. Math. Soc. 213 (1975), 217-235

8. D. W. Curtis and R. M. Schori, "Hyperspaces of Peano Continua are Hilbert cubes", Fund. Math. 101 (1978) 19-38

9. H. Torunczyk, "On CE-images of the Hilbert cube and characterization of Q-manifolds" Fund. Math. 106 (1980) 31-40

10. R. Duda, "On the hyperspace of subcontinua of a finite graph, I", Fund. Math. 62, (1968) 265-286

11. R. Duda, "On the hyperspace of subcontinua of a finite graph, II", Fund. Math. 63, (1968) 225-255

12. R. Duda, "Correction to the paper 'On the hyperspace of subcontinua of a finite graph, I' ", Fund. Math. 69, (1970) 207-211