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Abstract  

In this paper it will be constructed an abstract geometry will be called a triple space, which is defined in general sense by the closure theoretic definition of geometry “see [4]”. And it is proved that the category of triple spaces is isomorphic to the category of Steiner triple systems. And hence it could be shown that the class of Steiner triple systems which satisfy the geometric axiomI3,

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\forall x_1 ,x_2 ,x_{3,} y;ify \in< x_1 ,x_2 ,x_3 > \backslash< x_1 ,x_2 > \Rightarrow x_3 \in< x_1 ,x_3 ,y >$$ \end{document}
((I3)) is exactly the class of all Steiner triple systems in which every triangle generate a planar subsystem.