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- W2485626737 abstract "This chapter discusses the theory of cuttings of the plane. It discusses the auxiliary properties of polygonal arcs by presenting the proofs of the following theorems : (1) any two points of a connected open set R (i.e. of a region) situated on $$ 2 can be joined by a polygonal arc and (2) If L is a polygonal arc $$ $$ 2 , then the set $$2 - L is homeomorphic to the plane $$ 2 . The required homeomorphism h as follows: (1) on the exterior of the circle E 1 , h(z) = Z is set and (2) the annulus $$is mapped homeomorphically onto the annulus K1- K2i without affecting the values h on Fr ( E 1 ). The (closed or open) set A is called a cutting of the space $$ 2 if the set $$ 2 $$ A is not connected. The chapter also presents the proof of the theorem that states that if the closed set A cuts $$ 2 between p and q , then there exist two closed sets R and Q such that $$ = R ∪ Q , p ⊂ R, q $$ Q and R □ Q = A ." @default.
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- W2485626737 date "1972-01-01" @default.
- W2485626737 modified "2023-09-27" @default.
- W2485626737 title "CUTTINGS OF THE PLANE" @default.
- W2485626737 doi "https://doi.org/10.1016/b978-0-08-016160-0.50027-0" @default.
- W2485626737 hasPublicationYear "1972" @default.
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