A characterization of spaces \(l\)-equivalent to the unit interval.
First of all, let us quote the author's abstract: ``Two topological spaces \(X\) and \(Y\) are said to be \(\ell\)-equivalent if there is a linear homeomorphism between \(C_p(X)\) and \(C_p(Y)\), the spaces of continuous functions on \(X\) and \(Y\) endowed with the topology of pointwise convergence. We prove a necessary condition on a topological space to be \(\ell\)-equivalent to the \(n\)-dimensional cube \(I^n\).'' The author proves the following theorems. Definition 5. Let \(X\) be a topological space. For an \(n\in\mathbb{N}\), let \[ I_n(X)= \bigcup_{s\in S} U_s,\quad J_n(X)= \bigcup_{t\in T} V_t, \] where \(\{U_s : s\in S\}\) is the family of all open subsets of \(X\) which can be embedded into \(I^n\) and \(\{V_t: t\in T\}\) is the family of all open subsets of \(X\) which are at most \((n- 1)\)-dimensional. For an ordinal \(\alpha\), the \(\alpha\)th \(n\)-dimensional embedding derivative \(X^{[\alpha,n]}\) is defined in the following way: \[ \begin{gathered} X^{[0,n]}= X;\\ X^{[\alpha+ 1,n]}= X^{[\alpha,n]}\setminus I_n(X^{[\alpha, n]});\\ X^{[\alpha,n]}= \bigcap_{\beta< \alpha} X^{[\beta,n]}\text{ if \(\alpha\) is limit}.\end{gathered} \] Similarly, the \(\alpha\)th \(n\)-dimensional strong embedding derivative \(X^{\alpha,n}\) is defined in the following way: \[ \begin{gathered} X^{0,n}= X\setminus J_n(X);\\ X^{\alpha+ 1,n}= X^{\alpha, n}\setminus (I_n(X^{\alpha, n})\cup J_n(X^{\alpha,n}\setminus I_n(X^{\alpha, n})));\\ X^{\alpha, n}= \bigcup_{\beta< \alpha} X^{\beta, n}\setminus J_n\Biggl(\bigcap_{\beta< \alpha} X^{\beta, n}\Biggr)\text{ if \(\alpha\) is limit}.\end{gathered} \] Theorem 6. If \(X\) is \(\ell\)-equivalent to \(I^n\), then \(X^{m,n}=\emptyset\) for some \(m\in\mathbb{N}\). For \(n= 1\) and a one-dimensional compact metrizable \(X\), this condition is sufficient. Theorem 18. Let \(X\) be an \(n\)-dimensional compact metric space, and let \[ H: C_p(X)\to C_p(X) \] be a linear homeomorphism. If \(X^{m,n}= \phi\) for some \(m\in\mathbb{N}\), then \(L(H)\geq \sqrt{m}\), where \(L(H)= \max\{\| H\|,\| H^{-1}\|\}\). Corollary 22. Let \(X\) and \(Y\) be \(\ell\)-equivalent compact metric spaces. Then \(X^{m,1}= \phi\) for some \(m\in\mathbb{N}\) if and only if \(Y^{m',1}= \phi\) for some \(m'\in\mathbb{N}\). The author also obtains several results relating to the above topics and already known facts.
scientific article; zbMATH DE number 2056834
scientific article