Question

help with the following question about homepmprphism

Answer 1

\[S=\left\{ (x,y) \epsilon R ^{2} :x ^{2}+y ^{2} =1\right\}\]
Defin\[h:[0,2\pi )\rightarrow S \]e by \[h(\theta)= \left( \cos \theta , \sin \right)\]

Answer 2

prove that it is continuous and bijective ,but is not a homoemorphism

Answer 3

S it is the unit circle in the plane

Answer 4

am i going to be wrong if i can Let epsilon >0 there exist delta >0 such that whenever |x-a|<delta ,then |h(theta)-h(a)|=\[|\left( \cos(\theta) ,\sin(\theta)) - (\cos(a),\sin(a))\right)\]