We have calculated distributions and lifetimes of positrons in the infinite-layer compound SrCuO2 and those trapped at possible point defects therein. In the delocalized state, positrons show their density maxima at interstitial sites in the Sr planes and have a significant overlap also with Cu and O atoms. The corresponding positron lifetime is 149 ps. It has been revealed that the Sr vacancy strongly localizes positrons with the binding energy of 2.8 eV and the lifetime of 238 ps, while the O vacancy does not trap positrons. Calculations are also performed on related materials Sr2Cu4O6 and Sr4Cu6O10, which are characterized by one-dimensional networks of edge-sharing CuO4 squares. Positrons are predominantly distributed between these networks in these materials and their corresponding lifetimes are 170 171 ps.