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 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,
The property of compactness is extended to metric and, more generally, topological space. In a metric space a set is compact
if any sequence of its elements contains a convergent subsequence whose limit belongs to the set, or equivalently,
if the set is closed and bounded.
In a topological space, a set is compact
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,
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,
One way to embed R into a compact set is by adding two "end points" -∞ and +∞ and declaring their neighborhoods to be the sets
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,
References
- T. Gowers (ed.), The Princeton Companion to Mathematics, Princeton University Press, 2008
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