Final answer:
To prove that the 1 point compactification of R is homeomorphic to the circle, we need to show that there is a continuous and bijective mapping between the two spaces.
Step-by-step explanation:
To prove that the 1 point compactification of R is homeomorphic to the circle, we need to show that there is a continuous and bijective mapping between the two spaces. Let's consider the 1 point compactification of R, denoted as R*.
We can define a mapping f: R* -> Circle, where Circle represents the unit circle in the plane. We can map the points of R onto the unit circle by using a function such as f(x) = (cos(2*pi*x), sin(2*pi*x)), where x represents a point in R. This function is continuous and bijective, meaning it preserves the topology of R* and the unit circle, making them homeomorphic.