# Existence of infinitely many maximal ideals in …

Let $$\mathscr{A}_p=\{f\in C[0,1]:f(p)=0\}$$. Then we all know that every maximal ideal of $$C[0,1]$$ is of the form $$\mathscr{A}_p$$ for some $$p\in [0,1]$$.

Instead of considering compact set $$[0,1]$$, if we take an open set $$(0,1)$$, then $$\mathscr{M}_p=\{f\in C(0,1):f(p)=0\}$$ is also maximal ideal in $$C(0,1)$$ for $$p\in (0,1)$$.

But can we have a maximal ideal of $$C(0,1)$$ which is not of the form $$\mathscr{M}_p$$ for all $$p\in (0,1)$$ ? The answer is YES. I have constructed such ideal.

Let $$\mathscr{M}=\{f\in C(0,1):f\text{ is of compact support}\}$$. Clearly this is an ideal. Now take any $$p\in (0,1)$$. Then we can clearly construct a continuous function which is non-zero at point $$p$$ and has a compact support. Hence we have showed that $$\mathscr{M}\not= \mathscr{M}_p$$ for any $$p\in (0,1)$$. Thus we have a maximal ideal which is not of the form $$\mathscr{M}_p$$ for all $$p$$.

Now the $$\textbf{first question}$$ is : Can we have an infinite number of maximal ideals which are not of the form $$\mathscr{M}_p$$ ?

The $$\textbf{second question}$$ is : Consider the map $$\mathbb{R}\to C(0,1)$$ which sends $$a\in\mathbb{R}$$ to a constant function with value $$a$$. If $$\mathscr{N}$$ is a maximal ideal of $$C(0,1)$$ such that $$\mathbb{R}\longrightarrow C(0,1) \longrightarrow C(0,1)/\mathscr{N}$$ is an isomorphism, then show that $$\mathscr{N}$$ has to be $$\mathscr{M}_p$$ for some $$p\in (0,1)$$.

I don't know how to even start. Any help or hint will be appreciated. Thank you in advance.

• You probably meant $\mathscr{M}$ is not included in $\mathscr{M}_p$, not $\neq$ . – Max Feb 11 at 9:01