Fixed point sets of Tychonov cubes
The authors, applying advanced methods of infinite-dimensional topology, give a spectral characterization of compact subsets of a Tikhonov cube \(Q^\tau\), \(\tau \geq \omega_1\), which are the sets of fixed points of continuous maps of the cube \(Q^\tau\) into itself. One of the main results says that \(X\) is the set of fixed points of a continuous map \(f : Q^\tau \to Q^\tau\) if and only if \(X\) can be represented as the limit of an inverse system \(\{X_\alpha, p_\alpha^{\alpha + 1}, \alpha < \tau\}\) whose short projections \(p_\alpha^{\alpha + 1} : X_{\alpha + 1} \to X_\alpha\) are stable and whose first element \(X_0\) is metrizable. The paper is interesting in view of the following known facts: (1) [\textit{J. R. Martin}, Proc. Am. Math. Soc. 103, No. 4, 1293-1298 (1988; Zbl 0652.54029)] For each closed subset \(X\) of the Hilbert cube \(Q^\omega\) there exists a continuous map \(f:Q^\omega \to Q^\omega\) such that \(X = \{x : f(x) = x\}\). (2) [\textit{P. Koszmider}, Can. Math. Bull. 35, No. 2, 221-229 (1992; Zbl 0796.54056)] There exists a zero-dimensional closed subset of \(Q^\tau\), \(\tau \geq \omega_1\), which does not coincide with the set of fixed points of any continuous map \(f : Q^\tau \to Q^\tau\).
limits of inverse systems over well-ordered sets
scientific article; zbMATH DE number 792535
scientific article