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Let $R=\mathcal{C}(X,\mathbb{R})$ denote ring of all continuous real valued functions on $X(\subseteq \Bbb{R})$.

Let $X=[0,1]$. Then We know for any $\alpha\in X$, $M_{\alpha}=\{f\in R:f(\alpha)=0\}$ is a maximal ideal of $R$ and also, any maximal ideal of $R$ is $M_{\alpha}$ for some $\alpha\in X$.

But I got to know that the latter one of above does not hold if we take $X=(0,1)$. That means, there is a maximal ideal of $\mathcal{C}((0,1),\mathbb{R})$ which is not of the form $M_{\alpha}$.

Can anyone please give such an example of a maximal ideal of $\mathcal{C}((0,1),\mathbb{R})$. Thank you.

Avisek Sharma
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Hint: Let $I$ be the ideal of all continuous functions $f:(0,1)\to \Bbb R$ such that $f\big(\frac{1}{n}\big)=0$ for all but finitely many $n\in\Bbb N$. An example of such a function is $(0,1)\ni x\longmapsto \sin\big( \frac{\pi}{x}\big)\in \Bbb R$. Then, there is a maximal ideal $M$ containing $I$, as $I$ doesn't contains unit element of $\mathcal C\big((0,1),\Bbb R\big)$. See here every ideal is contained in a maximal ideal Note that this $M$ can't be of the above form.

The second paragraph has been deleted, as it has nothing to do with OP's question.

Generalization: Let $X$ be a non-compact topological space, so there is net $\{x_\alpha:\alpha\in \mathcal D\}$ having no convergent sub-net. Here, $\mathcal D$ is a directed set. Now, give a similar argument as in Hint.

Sumanta
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