To verify this, we just need to verify that the closure satisfies the appropriate universal property. In order to then get this for general compact Hausdorff K we use the above to note that K can be embedded in some cube, extend each of the coordinate functions and then take the product of these extensions. The special property of the unit interval needed for this construction to work is that it is a cogenerator of the category of compact Hausdorff spaces: this means that if A and B are compact Hausdorff spaces, and f and g are distinct maps from A to B , then there is a map h from B to [0, 1] such that hf and hg are distinct.
Any other cogenerator or cogenerating set can be used in this construction. This may be verified to be a continuous extension of f. This is really the same construction, as the Stone space of this Boolean algebra is the set of ultrafilters or equivalently prime ideals, or homomorphisms to the 2 element Boolean algebra of the Boolean algebra, which is the same as the set of ultrafilters on X.
The construction can be generalized to arbitrary Tychonoff spaces by using maximal filters of zero sets instead of ultrafilters. Notice that C b X is canonically isomorphic to the multiplier algebra of C 0 X. In the case where X is locally compact , e. The major results motivating this are Parovicenko's theorems , essentially characterising its behaviour under the assumption of the continuum hypothesis.
This comprehensive book is about the study of invariant pseudodistances non-negative functions on pairs of points and pseudometrics non-negative functions on the tangent bundle in several complex variables. Les mer. Om boka As in the field of "Invariant Distances and Metrics in Complex Analysis" there was and is a continuous progress this is now the second extended edition of the corresponding monograph. It is an overview over a highly active research area at the borderline between complex analysis, functional analysis and differential geometry.
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