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Infinity in Compactification
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
In a topological space, a set is compact
(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
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