Topological finiteness, Bernstein sets, and topological rigidity (Q408578)

In this paper, observing the fact that the finiteness of a set \(X\) is characterized by the property that there is no bijection from \(X\) onto a proper subset of \(X\), the author defines the notion of topological finiteness of a topological space \(X\) by the property that there is no homeomorphism from \(X\) onto a proper subset of \(X\). For example, the \(n\)-dimensional sphere \(S^n\) (\(n \geq 0\)) is topologically finite, and no \(n\)-dimensional subspace of \({\mathbb R}^n\) (\(n \geq 1\)) is topologically finite. NEWLINENEWLINENEWLINE NEWLINEThe main result of the paper is that for every \(0 \leq n \leq \infty\) there exists an \(n\)-dimensional separable and metrizable space \(X_n\) with \(|X_n| = \mathfrak{c}\) which is topologically finite but not strongly rigid. NEWLINEThe notion of strong rigidity is in the sense of \textit{J. J. Charatonik} [Houston J. Math. 26, 639--660 (2000; Zbl 0981.54017)]. A non-degenerate topological space \(X\) is said to be strongly rigid iff the only homeomorphism of \(X\) to \(X\) is the identity map. Every strongly rigid space is obviously topologically finite. The author also discusses some basic properties of topological finite spaces.