In semi-algebraic geometry examples of open morphisms occur naturally, for example coverings. This paper emphasizes the investigation of open morphisms of locally semi-algebraic spaces from the point of view of real closed spaces. This means that real spectra and algebraic properties of rings of sections of locally semi-algebraic spaces play a role. Typical examples for results coming up in this context are the following: Open morphisms can be characterized in terms of a lifting property of

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... ialization chains in real spectra. For morphisms into semi-algebraic paths, openness and flatness are equivalent conditions. Rg------+R c n cn " B-C commutes (where we identify Ii" Rc with the fields of representatives). We can write B=R,i-m" C= R,tm,.