I will first review some facts in Teichmüller theory of Riemann surfaces. After that I will define the Teichmüller space of a subset of the Riemann sphere. There is a natural Teichmüller metric from the definition. Since this space is a complex manifold, it has the Kobayashi metric. I will show these two metrics coincide by an outline of our proof of the lift theorem. If the subset is infinite, the Teichmüller space is infinite-dimensional. There may have more than one geodesic connecting two points in the space. I will talk some equivalence statements for the uniqueness based on Beltrami coefficients and holomorphic isometrics. This talk is based on the work jointly with my colleague Sudeb Mitra and my students Zhe Wang, Michael Beck, and Nishan Chatterjee at CUNY.