On the minimal cover property and certain notions of finite (Q1661686)

Let \(X\) be a topological space. A cover \(\mathcal U\) of \(X\) is said to be a minimal cover, if for each \(U \in \mathcal U\), the family \(\mathcal U \setminus \{ U \}\) is not a cover of \(X\). \(X\) has the minimal cover property (mcp), if for each cover \(\mathcal U\) of \(X\) there exists a minimal subcover \(\mathcal V \subseteq \mathcal V\). \textsf{MCC} is the following principle: Every topological space with mcp is compact. In \textsf{ZFC}, \textsf{MCC} is true. But this is not the case in \textsf{ZF}. It was shown in [\textit{P. Howard} and \textit{E. Tachtsis}, Topology Appl. 173, 94--106 (2014; Zbl 1305.03038)] that a topological space with mcp is countably compact. In this paper, the author investigates the deductive strength of the principle ``every topological space with mcp is compact''. This is done with respect of notions of finite as well as with properties of linearly ordered sets and partially ordered sets. These properties concern the existence of countably infinite subsets for infinitely ordered sets, the existence of infinite chains or infinite antichains in infinite partially ordered sets, the existence of cofinal well-founded subsets of partially ordered sets. The author works with Fraenkel-Mostowski permutation models and Pincus' transfer theorems.