In mathematics, the Federer–Morse theorem, introduced by , states that if ''f'' is a
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
from a
compact metric space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
''X'' to a compact metric space ''Y'', then there is a
Borel subset
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are name ...
''Z'' of ''X'' such that ''f'' restricted to ''Z'' is a
bijection from ''Z'' to ''Y''.
Moreover, the inverse of that restriction is a Borel
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sign ...
of ''f''—it is a
Borel isomorphism In mathematics, a Borel isomorphism is a measurable bijective function between two measurable standard Borel spaces. By Souslin's theorem in standard Borel spaces (a set that is both analytic and coanalytic is necessarily Borel), the inverse of ...
.
[Page 12 of ]
See also
*
Uniformization
*
Hahn–Banach theorem
The Hahn–Banach theorem is a central tool in functional analysis.
It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
References
*
*
*
*
Further reading
* L. W. Baggett and Arlan Ramsay, ''A Functional Analytic Proof of a Selection Lemma'', Can. J. Math., vol. XXXII, no 2, 1980, pp. 441–448.
{{DEFAULTSORT:Federer-Morse theorem
Theorems in topology