Continuous maps and homeomorphisms
General · Mathematics
Study notes
Q: Show (0,1) ≅ ℝ via homeomorphism. Is [0,1] ≅ (0,1)? f(x) = tan(π(x-1/2)): (0,1)→ℝ: bijective, continuous, inverse continuous! HOMEOMORPHISM: (0,1) ≅ ℝ! [0,1] vs (0,1): remove endpoint: [0,1]\{0} connected; (0,1)\{x} disconnected! Not homeomorphic (connectedness invariant!)!