I am learning hopf algebras by D.E Radford, questions about cofiniteness occur to me but I can't find out the answers.
My question is
Let $U$ be a vector space over a field $k$. Is there any subspace $I$ of $U^*$ such that $I$ is cofinite but not closed ?
The definition of closedness is given as below:
Suppose $V$ is a subset of $U$ and $X$ is a subset of $U^*$. Write $V^\perp = \{u^∗ \in U^∗ | u^∗(V ) = (0)\}$ and $X^\perp = \{u ∈ U | X(u) = (0)\}$.
A subspace X of $U^*$ is said to be closed if $X = X^{⊥⊥}$.
My quesiton came from some statements of closedness:
Corollary 1.3.9$\ $ Fnite dimensional subspaces of $U^*$ are closed.
Proposition 1.3.12$\ $ Let $I$,$J$ be cofinite subspaces of $U^*$. If $I$ is closed and $J ⊇ I$, then J is closed.
Prop 1.3.12 might suggest that cofinite subspaces needn't to be closed, but I can't think out an example.
Edit:$\ $ I restate the question since part of the original question is trivial.
If $I$,$J$ are cofinite subspaces of $U^*$, then $I\cap J$ is cofinite in $U^*$ ? Since $I\cap J$ is cofinite in $I$ and cofiniteness is transmissable, the answer is trivial.