Spaces without nonconstant maps into \(Y\) (Q1840744)

Given a topological space \(Y\), a topological space \(X\) is called \(Y\)-connected if every continuous map \(X\to Y\) is constant. It is well known that the category \(\mathcal H(Y)\) of nonvoid \(Y\)-connected regular \(T_1\)-spaces is always rather complex. The paper brings further results in this direction: (a) \(\mathcal H(Y)\) always contains two reflective subcategories with a nonreflective intersection and (b) \(\mathcal H(Y)\) contains arbitrarily large extremally semirigid spaces.