Infinity in Compactification

... let me mention the emergence of a remarkably useful and natural concept: that of compact sets. These occurred first among other classes of sets with somewhat different definitions. Eventually the modern concept of a compact set was singled out as being the "right" one.

David Ruelle
The Mathematician's Brain
Princeton University Press, 2008, p. 75

The term compact set has been introduced by Maurice René Fréchet in his 1906 Ph.D. thesis. Compact sets have many useful properties but the most essential I believe was distilled from a property of closed bounded segments of a straight line, [a, b]. Any continuous function on such a segment attains its extreme values, maximum and minimum. And the same (the Extreme Value Theorem) holds for all compact sets.

The property of compactness is extended to metric and, more generally, topological space. In a metric space a set is compact

  1. if any sequence of its elements contains a convergent subsequence whose limit belongs to the set, or equivalently,

  2. if the set is closed and bounded.

In a topological space, a set is compact

  1. if it possesses the Finite Intersection Property, i.e. if, for every collection of closed subsets whose finite subcollections have non-empty intersection, the intersection of the whole collection is also non-empty. Equivalently,

  2. if it possesses the Finite Cover Property: in any collection of open sets whose union contains the given set, there is a finite subcollection whose union still covers the set.

(Let's remark in passing that the Finite Intersection Property plays an important role in a ralization of the Non-Standard Analysis in the form of Compactness Theorem.)

In a topological space, the latter two properties are obviously equivalent which can be seen by passing from a set to its complement. Sets that posses either are often called bicompact. In a metric space, all four properties are equivalent.

Every topological space can be embedded into a compact one, and this in a variety of ways. Let's consider, for example, the straight line R with the usual topology induced by the standard metric, d(x, y) = |x - y|. R is not compact because, for one, there is a sequence, say a sequence of positive integers, that contains no convergent subsequence. There is also a cover by open intervals (n - 1, n + 1), n = 0, 1, 2, ... that does not contain a finite subcover.

One way to embed R into a compact set is by adding two "end points" -∞ and +∞ and declaring their neighborhoods to be the sets {x: x < n} and {x: x > n}, respectively, where n is any real number. The resulting set will be very much like a finite closed interval, now [-∞, +∞] and will be compact because any cover will include (at least) two sets - one covering +∞, the other -∞ and the remaining cover a compact set (a closed finite portion of R.)

Such an embedding of a topological space X into a compact space C is called compactification. R admits a prototypical one point, or Alexandroff's, compactification R. R is enlarged by adding a single point ∞ whose neighborhoods are defined as complements of compact subsets of R.

Topologically, this transforms the real line into a circle with naturally induced by the topology of the plane.

It's noteworthy, that in both variants of compactification the sequence of positive integers {n} becomes convergent, to boot, limn→∞n = ∞. For the two point compactification, limn→-∞n = -∞, whereas, for the one point compactification, limn→∞n = ∞, regardless, in a sense, how n moves to infinity - left or right. In particular, the sequence 1, -2, 3, -4, 5, ... converges to ∞.

References

  1. T. Gowers (ed.), The Princeton Companion to Mathematics, Princeton University Press, 2008

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