Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4658263 | Topology and its Applications | 2015 | 11 Pages |
Abstract
We prove that if Y is a continuum containing a solenoid Σ such that R=Y∖ΣR=Y∖Σ is homeomorphic to [0,∞)[0,∞), then, there exists a retraction r of Y to Σ. Moreover, any two such retractions are homotopic. It follows that if PRPR is the arc component of Σ containing r(R)r(R), then PRPR is invariant under every homeomorphism of Y into itself.We also prove the following theorem for Knaster continua. Suppose Y is a continuum containing a Knaster continuum K such that R=Y∖KR=Y∖K is homeomorphic to [0,∞)[0,∞) and cl(R)=Y. Let e denote the endpoint of R and let x∈Kx∈K. Then there exists a retraction r of Y to K such that r(e)=xr(e)=x.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Geometry and Topology
Authors
Piotr Minc,