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\begin{document}$${\mathcal {C}}$$\end{document} be a class of topological semigroups. A semigroup X is called absolutely C\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}$$\end{document}-closed if for any homomorphism h:X→Y\documentclass[12pt]{minimal}
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\begin{document}$$h:X\rightarrow Y$$\end{document} to a topological semigroup Y∈C\documentclass[12pt]{minimal}
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\begin{document}$$Y\in {\mathcal {C}}$$\end{document}, the image h[X] is closed in Y. Let T1S\documentclass[12pt]{minimal}
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\begin{document}$$\textsf {T}_{\!\textsf {1}}\textsf {S}$$\end{document}, T2S\documentclass[12pt]{minimal}
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\begin{document}$$\textsf {T}_{\!\textsf {2}}\textsf {S}$$\end{document}, and TzS\documentclass[12pt]{minimal}
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\begin{document}$$\textsf {T}_{\!\textsf {z}}\textsf {S}$$\end{document} be the classes of T1\documentclass[12pt]{minimal}
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\begin{document}$$T_1$$\end{document}, Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup X is absolutely TzS\documentclass[12pt]{minimal}
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\begin{document}$$\textsf {T}_{\!\textsf {z}}\textsf {S}$$\end{document}-closed if and only if X is absolutely T2S\documentclass[12pt]{minimal}
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\begin{document}$$\textsf {T}_{\!\textsf {2}}\textsf {S}$$\end{document}-closed if and only if X is chain-finite, bounded, group-finite and Clifford + finite. On the other hand, a commutative semigroup X is absolutely T1S\documentclass[12pt]{minimal}
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\begin{document}$$\textsf {T}_{\!\textsf {1}}\textsf {S}$$\end{document}-closed if and only if X is finite. Also, for a given absolutely C\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}$$\end{document}-closed semigroup X we detect absolutely C\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}$$\end{document}-closed subsemigroups in the center of X.
