An Elementary Proof of the Jordan-schoenflies Theorem1
نویسنده
چکیده
The exterior of a bounded closed point set b in E will mean the unbounded region of the complementary set E — b. The remainder of E — b, if not vacuous, will be called the interior of b. (A) As a corollary to the above theorem, c is intersected by any simple arc with one end point interior and one exterior to c. This paper contains an elementary constructive proof of the Jordan-Schoenflies Theorem, motivated by the belief that such a proof should be presented at a fairly early stage to students of topology and analysis. To that end, it is desirable that the argument be disassociated from conformai mapping theory and be accomplished by methods as elementary as possible.
منابع مشابه
Classifying Simple Closed Curve Pairs in the 2-sphere and a Generalization of the Schoenflies Theorem1
A classification theory is developed for pairs of simple closed curves (A,B) in the sphere S, assuming that A ∩ B has finitely many components. Such a pair of simple closed curves is called an SCC-pair, and two SCC-pairs (A,B) and (A′, B′) are equivalent if there is a homeomorphism from S to itself sending A to A′ and B to B′. The simple cases where A and B coincide or A and B are disjoint are ...
متن کاملOn p-semilinear transformations
In this paper, we introduce $p$-semilinear transformations for linear algebras over a field ${bf F}$ of positive characteristic $p$, discuss initially the elementary properties of $p$-semilinear transformations, make use of it to give some characterizations of linear algebras over a field ${bf F}$ of positive characteristic $p$. Moreover, we find a one-to-one correspondence between $p$-semiline...
متن کاملA new proof for the Banach-Zarecki theorem: A light on integrability and continuity
To demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the Banach-Zareckitheorem is presented on the basis of the Radon-Nikodym theoremwhich emphasizes on measure-type properties of the Lebesgueintegral. The Banach-Zarecki theorem says that a real-valuedfunction $F$ is absolutely continuous on a finite closed intervalif and only if it is continuo...
متن کاملOn Shyr-Yu Theorem1
An alternative proof of Shyr-Yu Theorem is given. Some generalizations are also considered using fractional root decompositions and fractional exponents of words.
متن کاملEla the Jordan
The relationship between the Jordan forms of the matrix products AB and BA for some given A and B was first described by Harley Flanders in 1951. Their non-zero eigenvalues and non-singular Jordan structures are the same, but their singular Jordan block sizes can differ by 1. We present an elementary proof that owes its simplicity to a novel use of the Weyr characteristic.
متن کامل