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Given an algebraic function field $F|K$ and a place $\wp$ on $K$, we provethat the places that are composite with extensions of $\wp$ to finiteextensions of $K$ lie dense in the space of all places of $F$, in a strongsense. We apply the result to the case of $K=R$ any real closed field and thefixed place on $R$ being its natural (finest) real place. This leads to a newdescription of the real holomorphy ring of $F$ which can be seen as an analogueto a certain refinement of Artin's solution of Hilbert's 17th problem. We alsodetermine the relation between the topological space $M(F)$ of all $\R$-placesof $F$ (places with residue field contained in $\R$), its subspace of all$\R$-places of $F$ that are composite with the natural $\R$-place of $R$, andthe topological space of all $R$-rational places. Further results about thesespaces as well as various classes of relative real holomorphy rings are proven.At the conclusion of the paper the theory of real spectra of rings will beapplied to interpret basic concepts from that angle and to show that the space$M(F)$ has only finitely many topological components.

Density of Composite Places in Function Fields and Applications to Real Holomorphy Rings