In this paper, we consider the complementarity of the space $C_p(X)$ in the space $C_p(Y)$ for countable sparse metrizable spaces $X$ and $Y$. It is said that the space $C_p(X)$ is complementably embedded in the space $C_p(Y)$ if there exists a linear homeomorphism $C_p(X)$ to the complemented subspace $L\subset C_p(Y)$. We prove that if for some ordinal $\alpha$ the derivative $X^{(\alpha\cdot\omega)}\neq\varnothing$, and $Y^{(\omega)}=\varnothing$, then the space $C_p(X)$ is complementably not embedded in the space $C_p(Y)$. We also consider the derivatives $X^{(\alpha)}$, which are defined similarly to $X^{(\alpha)}$ by removing all points having a compact neighborhood. It is proved that if $X^{\{\alpha\}}\neq\varnothing$, and $Y^{\{\alpha\}}=\varnothing$, then the space $C_p(X)$ is not complementably embedded in the space $C_p(Y)$. Futhermore, if $X^{\{\alpha\}}=Y^{\{\alpha\}}=\varnothing$, $X^{\{\alpha-1\}}$ is a locally compact non-compact space, and $Y^{\{\alpha -1\}}$ is compact, then the space $C_p(X)$ is complementably not embedded in the space $C_p(Y)$. For the proof, the method of decomposition of the space $C_p(X)$ into a countable product of the spaces $C_p(X_n)$ and the existence of a continuous linear extension operator $T:C_p(L)\longrightarrow C_p(X)$ for a closed subset of $L\subset X$.
Keywords: homeomorphism, linear homeomorphism, topology of pointwise convergence, retract, projector, complemented subspaces, ordinal, closed graph theorem
Received November 27, 2024
Revised February 14, 2025
Accepted February 17, 2025
Tatyana Khmyleva, Cand. Sci. (Phys.-Math.), the Department of Mathematical Analysis and Theory of Functions, Tomsk State University, 634050 Russia, e-mail: tex2150@yandex.ru
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Cite this article as: T.E. Khmyleva. On complementarity and linear homeomorphism of $C_p(X)$ spaces for countable metric spaces $X$. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 1, pp. 236–246.
