I have studied about maximal ideals in $C[0,1]$ . They are precisely of the form $$\{f\in C[0,1] | f(c)=0\} \text{ for } c\in [0,1].$$

If we replace $[0,1]$ by $(0,1)$ and then look at $C(0,1)$, then obviously $M_c = \{f\in C(0,1) | f(c)=0\}$ are maximal ideals in $C(0,1)$ , $\forall c \in (0,1)$ , but since we do not have the compactness of $[0,1]$ anymore, I guess there are some other maximal ideals as well.

I am trying to solve the problem by first giving an existential argument and then by showing an explicit maximal ideal other that of the form $M_c$ for some $c \in (0,1)$ .

Rings of Continuous Functions. (I guess somebody who know more about this will post a more detailed answer soon.) However, I am not sure whether this is too helpful, since Stone-Cech compactification of $(0,1)$ seems like a quite complicated object. $\endgroup$ – Martin Sleziak Apr 27 '18 at 12:213more comments