Spaces in which countable closed sets have countable character (Q5932741)

A topological space \(X\) is called a \(CC_\omega\)-space if every countable closed subset of \(X\) has countable character. It is proved that a \(T_3\)-space \(X\) is a \(CC_\omega\)-space if and only if the set of limit points of \(X\) is countably compact and every compact subset has countable character. It is also shown that among \(T_3\)-spaces, \(CC_\omega\)-spaces are \(WN\)-spaces and are very close to \(M\)-spaces.