A property similar to countable tightness I am interested in topological spaces having the following property: > A function $f\colon X\to \mathbb R$ is continuous if and only if the restriction $f|_C$ is continuous for every countable subspace $C$ of $X$. What are such spaces called? Have they been studied? I'd be grateful for any pointers. * * * This property is, to some extent, similar to countable tightness. If the above condition would characterize continuity of functions from $X$ to arbitrary space $Y$ instead of $\mathbb R$, then we would get spaces with countable tightness; also known as countably generated spaces.

I think I have seen this in either McCoy, Ntantu: topological properties of spaces of continuous functions, and/or in Arhangel'skij's book on $C_p(X)$, the latter calls it $t_\mathbb{R}$ as a cardinal function IIRC. Something like real-tightness. So $t_{\mathbb{R}}(X) \le \tau$ iff ($f: X \rightarrow \mathbb{R}$ is continuous iff its restriction to all subsets of cardinality $\le \tau$ of $X$ is continuous).

I think it corresponds to some cardinal invariant of $C_p(X)$ and that this was the reason for its introduction. I don't have access to these books now, so I cannot check exactly, but I hope this helps anyway.