On complementarity and linear homeomorphism of $C_p(X)$ spaces for countable metric spaces $ X$
T. E. Khmyleva Tomsk State University, Faculty of Mechanics and Mathematics
Abstract:
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.
UDC:
517.977
MSC: 46E15,
46E40,
54C30 Received: 27.11.2024
Revised: 14.02.2025
Accepted: 17.02.2025
DOI:
10.21538/0134-4889-2025-31-1-236-246
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