 Logarithmic spiral as an embedded real line

The map $h$ is a homeomorphism from the real line to its image with the subspace topology: its continuous inverse is $h^{-1}(x,y)=\log \sqrt{x^2+y^2}$. Different parameters have different radii, so winding does not destroy injectivity. The origin is an accumulation point of the spiral but is not a point of this space.