On boundedly compact metrics and UC metrics
The author introduces the concepts of independent and strongly independent properties of metric spaces. Let \(M\) be the class of all metric spaces and let \(P_1\), \(P_2\) be some properties of metric spaces. Write \(M_i\) for the subclass of \(M\) of spaces satisfying property \(P_i\), \[ M_i = \{(X, d) \in M \colon (X, d)\text{ has the property } P_i\}, \quad i = 1, 2. \] The properties \(P_1\) and \(P_2\) are independent if \(M_1 \cap M_2\), \(M_1 \setminus M_2\), \(M_2 \setminus M_1\) and \(M \setminus (M_1 \cup M_2)\) are nonvoid subclasses of \(M\). Let \((X, \tau)\) be a metrizable topological space and let \(M(\tau)\) be the set of all metric spaces \((X, d)\) for which \(d \colon X^2 \to \mathbb{R}\) is compatible with \(\tau\). Write \(M_i(\tau) = M_i \cap M(\tau)\), \(i = 1\), \(2\). \textbf{Definition}. The properties \(P_1\) and \(P_2\) are strongly independent if the following implications \[ \bigl(M_1(\tau) \neq \varnothing \neq M_2(\tau)\bigr) \Rightarrow \bigl(M_1(\tau) \cap M_2(\tau) \neq \varnothing\bigr), \] \[ \bigl(M_1(\tau) \neq \varnothing \neq M(\tau) \setminus M_2(\tau)\bigr) \Rightarrow \bigl(M_1(\tau) \cap (M(\tau) \setminus M_2(\tau)) \neq \varnothing\bigr), \] \[ \bigl(M_2(\tau) \neq \varnothing \neq M(\tau) \setminus M_1(\tau)\bigr) \Rightarrow \bigl(M_2(\tau) \cap (M(\tau) \setminus M_1(\tau)) \neq \varnothing\bigr), \] \[ \bigl(M(\tau) \setminus M_1(\tau) \neq \varnothing \neq M(\tau) \setminus M_2(\tau)\bigr) \Rightarrow \bigl(M(\tau) \setminus (M_1(\tau) \cup M_2(\tau)) \neq \varnothing\bigr) \] are valid for every metrizable topological space \((X, \tau)\). Recall that a metric space \((X, d)\) is boundedly compact if every closed bounded \(A \subseteq X\) is compact. Moreover, a metric space is an UC space (= Atsuji space) if every continuous \(f \colon Y \to \mathbb{R}\) is uniformly continuous. It is proved that boundedly compactness and UC-ness form a strongly independent pair of properties. Some other examples of independent pairs of properties of metric spaces are also given. It should be noted here that the concepts of independent properties and strongly independent ones seem to be attractive and, potentially, the future development of these concepts could lead to deep applications.
strongly independent properties of metrics
scientific article; zbMATH DE number 7324951
scientific article