SUMMARY
The discussion addresses the homeomorphism between a cylinder and a punctured plane, specifically $\mathbb{R}^2\setminus\{(0,0)\}$ and the cylinder defined as $C=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2=1\}$. An explicit homeomorphism is constructed using the function $$f:\mathbb{R}^2\setminus\{(0,0)\}\to C\;,\quad f(r\cos\theta,r\sin\theta)=(\cos \theta,\sin\theta,\ln r)\;(r>0)$$. The discussion concludes that while the punctured plane is homeomorphic to the cylinder, the full plane $\mathbb{R}^2$ is not homeomorphic to the punctured plane due to the property of simple connectivity.
PREREQUISITES
- Understanding of homeomorphism in topology
- Familiarity with punctured planes and their properties
- Knowledge of cylindrical coordinates
- Basic concepts of simple connectivity
NEXT STEPS
- Study the properties of homeomorphisms in topology
- Explore the concept of simple connectivity in more detail
- Learn about cylindrical coordinates and their applications
- Investigate other examples of homeomorphic spaces in topology
USEFUL FOR
Mathematicians, students of topology, and anyone interested in understanding the relationships between different geometric structures.