For infinite discrete topological space Y,\documentclass[12pt]{minimal}
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\begin{document}$$Y,$$\end{document} suppose A(Y)\documentclass[12pt]{minimal}
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\begin{document}$$A(Y)$$\end{document} is one point compactification of Y,\documentclass[12pt]{minimal}
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\begin{document}$$Y,$$\end{document} in the following text we prove that the transformation semigroup (A(Y),S)\documentclass[12pt]{minimal}
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\begin{document}$$(A(Y),S)$$\end{document} is distal if and only if the enveloping semigroup E(A(Y),S)\documentclass[12pt]{minimal}
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\begin{document}$$E(A(Y),S)$$\end{document} is a group of homeomorphisms on A(Y),\documentclass[12pt]{minimal}
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\begin{document}$$A(Y),$$\end{document} or equivalently for all p∈E(A(Y),S)\documentclass[12pt]{minimal}
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\begin{document}$$p \in E(A(Y),S)$$\end{document}, p:A(Y)→A(Y)\documentclass[12pt]{minimal}
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\begin{document}$$p:A(Y) \to A(Y)$$\end{document} is pointwise periodic. Also, the transformation group (A(Y),S)\documentclass[12pt]{minimal}
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\begin{document}$$(A(Y),S)$$\end{document} is distal (resp. equicontinuous, pointwise minimal) if and only if for all x∈A(Y)\documentclass[12pt]{minimal}
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\begin{document}$$x \in A(Y)$$\end{document}, xS\documentclass[12pt]{minimal}
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\begin{document}$$xS$$\end{document} is a finite subset of A(Y)\documentclass[12pt]{minimal}
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\begin{document}$$A(Y)$$\end{document}. The text is motivated with tables, counterexamples and studying finally distality (and co-decomposability to distal transformation semigroups) in the abelian transformation semigroup (A(Y),S)\documentclass[12pt]{minimal}
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\begin{document}$$(A(Y),S)$$\end{document}.
