We denote by $C_s(X)$ the set $C(X)$ of real-valued continuous functions defined on X endowed with the topology of the uniform convergence on the closed separable subspaces of X. In this paper we continue the study of $C_s(X)$ initiated in Pseudouniform topologies on $C(X)$ given by ideals (Pichardo-Mendoza et al. (2013) [7]). We prove that $C_s(X)$ is a k-space if and only if $C_s(X)$ is metrizable, and that compactness, sequential compactness and countable compactness coincide in subspaces of $C_s(X)$. In addition, we study the cellularity, density, weight and character of $C_s(X)$. We prove that (1) $d({({\mathbb{R}}^{2^{\lambda }})}_s)=\lambda$ if $\lambda ={\lambda }^{\omega }$, and (2) the Continuum Hypothesis is equivalent to the statement: Every non-separable space X satisfies $\chi (C_s(X))=w(C_s(X))$.

我们表示为 $C_{秒}(X)$ 集合 $C(X)$对定义的真实值连续函数X赋有一致收敛上的闭合可分离子空间拓扑X。在本文中，我们继续研究$C_{秒}(X)$在伪均匀拓扑中启动 $C(X)$ 由理想给出（Pichardo-Mendoza et al. (2013) [7]）。我们证明$C_{秒}(X)$是k空间当且仅当$C_{秒}(X)$ 是可度量的，并且紧致性、顺序紧致性和可数紧致性在子空间中重合 $C_{秒}(X)$. 此外，我们研究了细胞结构、密度、重量和特征$C_{秒}(X)$. 我们证明 (1)$d({({\mathbb{电阻}}^{2^{\lambda }})}_{秒})=\lambda$ 如果 $\lambda ={\lambda }^{\omega }$, (2) Continuum Hypothesis 等价于以下陈述：每个不可分空间X满足$\chi (C_{秒}(X))=瓦(C_{秒}(X))$.