The main purpose of this work is to study the metrizability of the space of R-places of a function field, with transcendence degree 1 over a real closed field. Specifically, we are going to present a necessary and sufficient condition that guarantees the metrizability of M (R(X )). In the first chapter, we give some preliminaries results on the topics of valuation, orderings, and real places. Then, in Chapter 2, we start with a straightforward approach to the problem. In this chapter, we will study the X (R(X )) and M (R(X )) spaces from a topological point of view.