Let P be a topological property. We say that a space X is dually P if for any neighbourhood assignment \({\mathcal {O}}=\{O_x: x\in X\}\) of X, there is a subspace \(Y \subset X\) with P such that \({\mathcal {O}}(Y)=X\). In this paper, we make several observations on dually (\({\mathfrak {c}}\)-)separable spaces. In particular, by using the idea of Buzyakova et al. (Comment Math Univ Carolin, 48(4): 689–697, 2007), we prove that every monotonically normal and \(\omega _1\)-weakly linearly Lindelöf space of countable tightness is dually separable, and hence it is Lindelöf. We also prove that every dually separable and monolithic space has countable extent. Finally, we prove that every dually \(\mathfrak c\)-separable space with a rank 2-diagonal has cardinality not exceeding \({\mathfrak {c}}\).

令P为拓扑性质。我们说一个空间X是双重P如果任何邻居分配\（{\ mathcal {Ø}} = \ {O_x：X \在X \} \）的X，有一个子空间\（Y \子集X \ ）与P使得\（{\ mathcal {O}}（Y）= X \）。在本文中，我们对双（\（{\ mathfrak {c}} \） -）可分离空间进行了观察。特别是，通过使用Buzyakova等人的想法。（评论数学大学Carolin，48（4）：689–697，2007年），我们证明每个单调正态和\（\ omega _1 \）-可数紧密度的弱线性Lindelöf空间是双重可分离的，因此它是Lindelöf。我们还证明，每个双重可分离且整体的空间都具有可数范围。最后，我们证明每个具有对角线2对角线的对偶\（\ mathfrak c \）可分离空间的基数不超过\（{\ mathfrak {c}} \）。