  Let $R$ be a reduced affine $\mathbb C$-algebra, with corresponding affine
algebraic set $X$. Let $\mathcal C(X)$ be the ring of continuous (Euclidean
topology) $\mathbb C$-valued functions on $X$. Brenner defined the
\emph{continuous closure} $I^{\rm cont}$ of an ideal $I$ as $I\mathcal C(X)
\cap R$. He also introduced an algebraic notion of \emph{axes closure} $I^{\rm
ax}$ that always contains $I^{\rm cont}$, and asked whether they coincide. We
extend the notion of axes closure to general Noetherian rings, defining $f \in
I^{\rm ax}$ if its image is in $IS$ for every homomorphism $R \to S$, where $S$
is a one-dimensional complete seminormal local ring. We also introduce the
\emph{natural closure} $I^\natural$ of $I$. One of many characterizations is
$I^\natural = I + \{f \in R: \exists n >0 \text{ with } f^n \in I^{n+1}\}$. We
show that $I^\natural \subseteq I^{\rm ax}$, and that when continuous closure
is defined, $I^\natural \subseteq I^{\rm cont }\subseteq I^{\rm ax}$. Under
mild hypotheses on the ring, we show that $I^\natural= I^{\rm ax}$ when $I$ is
primary to a maximal ideal, and that if $I$ has no embedded primes, then $I =
I^\natural$ if and only if $I = I^{\rm ax}$, so that $I^{\rm cont}$ agrees as
well. We deduce that in the polynomial ring $\mathbb C[x_1, \ldots, x_n]$, if
$f = 0$ at all points where all of the ${\partial f \over \partial x_i}$ are 0,
then $f \in ( {\partial f \over \partial x_1}, \, \ldots, \, {\partial f \over
\partial x_n})R$. We characterize $I^{\rm cont}$ for monomial ideals in
polynomial rings over $\mathbb C$, but we show that the inequalities
$I^\natural \subset I^{\rm cont}$ and $I^{\rm cont} \subset I^{\rm ax}$ can be
strict for monomial ideals even in dimension 3. Thus, $I^{\rm cont}$ and
$I^{\rm ax}$ need not agree, although we prove they are equal in $\mathbb
C[x_1, x_2]$.
