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A topological space  X  is  Baire  if the intersection of any sequence of open dense subsets of  X  is dense in  X . In this paper, we establish that the property ( κ ) for a Tychonoff space  X  is equivalent to Baireness of  B  1 ( X ), and hence, the Banakh property for  C   p  ( X ) is equivalent to meagerness of  B  1 ( X ). Thus, we obtain one characteristic of the Banakh property for  C   p  ( X ) through the property of the space  X .

The κ -Fréchet-Urysohn property for C p ( X ) is equivalent to baireness of B 1 ( X )