Preservation of Point Properties Under Refinable Maps

Determine whether each of the continuum properties \(\mathcal{P}_2\), \(\mathcal{P}_3\), \(\mathcal{P}_5\), and \(\mathcal{P}_6\) is preserved under refinable maps, where \(\mathcal{P}_i\) means that every point has the corresponding point property \(\mathbf{P}_i\): non-weak cut point, non-block point, non-strong center point, or non-cut point, respectively.

Background

The paper studies preservation of several global continuum properties under refinable maps. It establishes that $\mathcal{P}_4$, the property that every point is a shore point, is preserved under refinable maps, and recalls that $\mathcal{P}_1$, colocal connectedness, is also preserved. The corresponding preservation question remains unresolved for the intermediate and terminal properties $\mathcal{P}_2$, $\mathcal{P}_3$, $\mathcal{P}_5$, and $\mathcal{P}_6$.