  We prove that Thompson's group $F$ has a subgroup $H$ such that the conjugacy
problem in $H$ is undecidable and the membership problem in $H$ is easily
decidable. The subgroup $H$ of $F$ is a closed subgroup of $F$. That is, every
function in $F$ which is a piecewise-$H$ function belongs to $H$. Other
interesting examples of closed subgroups of $F$ include Jones' subgroups
$\overrightarrow{F}_n$ and Jones' $3$-colorable subgroup $\mathcal F$. By a
recent result of the first author, all maximal subgroups of $F$ of infinite
index are closed. In this paper we prove that if $K\leq F$ is finitely
generated then the closure of $K$, i.e., the smallest closed subgroup of $F$
which contains $K$, is finitely generated. We also prove that all finitely
generated closed subgroups of $F$ are undistorted in $F$. In particular, all
finitely generated maximal subgroups of $F$ are undistorted in $F$.
