Let $F$ be the power series ring in finitely many variables with coefficients in a field. let $\mathfrak{m}$ be the maximal ideal of $F$ generated by variables. Equip $F$ with the $\mathfrak{m}$-adic topology. My question is that is all ideals of $F$ closed?
Answer: From the general theory of noetherianccommutative algebras (see Chapter 10 of the Atiyah-Macdonald book [Introduction to commutative algerba]), we have $\cap_{r\geq 0} (I+\mathfrak{m}^r) = I$ for any ideal $I$ of $F$. It follows immediately that every ideal of $F$ is closed with respect to the $\mathfrak{m}$-adic topology.