The one-point compactification of a topological space is a new compact space obtained by adding a single new point “” to the original space and declaring in the complements of the original closed compact subspaces to be open.
One may think of the new point added as the “point at infinity” of the original space. A continuous function on vanishes at infinity precisely if it extends to a continuous function on and literally takes the value zero at the point “”.
This one-point compactification is also known as the Alexandroff compactification after a 1924 paper by Павел Сергеевич Александров (then transliterated ‘P.S. Aleksandroff’).
The one-point compactification is usually applied to a non-compact locally compact Hausdorff space. In the more general situation, it may not really be a compactification and hence is called the one-point extension or Alexandroff extension.
For topological spaces
Let be any topological space. Its one-point extension is the topological space
(If is Hausdorff, then its compact subsets must always be closed, so (2) is often given in a simpler form.)
For non-commutative topological spaces (-algebras)
Dually in non-commutative topology the one-point compactification corresponds to the unitisation of C*-algebras.
The evident inclusion is an open embedding.
The one-point compactification is universal among all compact spaces into which has an open embedding, so it is essentially unique.
is dense in iff is not already compact. Note that is technically a compactification of only in this case.
is Hausdorff (hence a compactum) if and only if is already both Hausdorff and locally compact.
The operation of one-point compactification is not a functor on the whole category of topological spaces. But it does extend to a functor on topological spaces with proper maps between them.
For the -sphere (as a topological space) is the one-point compactification of the Cartesian space
Via this presentation of the -sphere the canonical action of the orthogonal group on induces an action of on , which preserves the basepoint (the “point at infinity”).
This construction presents the J-homomorphism in stable homotopy theory and is encoded for instance in the definition of orthogonal spectra.
Slightly more generally, for any real vector space of dimension one has . In this context and in view of the previous case, one usually writes
for the -sphere obtained as the one-point compactification of the vector space .
For a compact topological space and a vector bundle, then the (homotopy type of the) one-point compactification of the total space is the Thom space of , equivalent to .
For a simple example: the real projective plane is the one-point compactification of the ‘open’ Möbius strip, as line bundle over . This is a special case of the more general observation that is the Thom space of the tautological line bundle? over .
- John Kelly, General Topology (1975)