Recently I have been studying Rings of real-valued continuous functions. I came up with the following question that I am not able to answer. Please help me.
Does there exist a topological space $X$ and a point $x_{0} \in X$ such that $C(X) - (M_{x_{0}} \cup T)$ is countably infinite? (here, $M_{x_{0}} = \{f \in C(X) \mid f(x_{0}) = 0\}$ and $T$ is the set of all constant functions from $X$ to $\mathbb{R}$.)