Continuous closure, axes closure, and natural closure

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} $I^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}$ of an ideal $I$ as $I\mathcal C(X) \cap R$\mathrm{.}Healsointroducedanalgebraicnotionof\mathit{\text{axes}}\mathit{\text{closure}}$I^{\rm ax}$thatalwayscontains$I^{\rm cont}$, and asked whether they coincide. We extend the notion of axes closure to general Noetherian rings, defining $f \in I^{\rm ax}$ifitsimageisin$IS$foreveryhomomorphism$R \to S$,where$S$ is a one-dimensional complete seminormal local ring. We also introduce the \emph{natural closure} $I^{\mathrm{♮}}$ of $I$. One of many characterizations is $I^{\mathrm{♮}}=I+\{f\in R:\mathrm{\exists }n>0\text{ with }{f}^n\in I^{n+1}\}$. We show that $I^{\mathrm{♮}}\subseteq I^{\mathrm{a}\mathrm{x}}$, and that when continuous closure is defined, $I^{\mathrm{♮}}\subseteq I^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}\subseteq I^{\mathrm{a}\mathrm{x}}$. Under mild hypotheses on the ring, we show that $I^{\mathrm{♮}}=I^{\mathrm{a}\mathrm{x}}$ when $I$ is primary to a maximal ideal, and that if $I$ has no embedded primes, then $I = I^\natural$ifandonlyif$I = I^{\rm ax}$,sothat$I^{\rm cont}$ agrees as well. We deduce that in the polynomial ring $\mathbb{C}[x_1,\dots ,x_n]$, if $f=0$ at all points where all of the $\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}_i}$ are 0, then $f \in ( {\partial f \over \partial x_1}, , \ldots, , {\partial f \over \partial x_n})R$\mathrm{.}Wecharacterize$I^{\rm cont}$ for monomial ideals in polynomial rings over $\mathbb{C}$, but we show that the inequalities $I^{\mathrm{♮}}\subset I^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}$ and $I^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}\subset I^{\mathrm{a}\mathrm{x}}$ can be strict for monomial ideals even in dimension 3. Thus, $I^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}$ and $I^{\mathrm{a}\mathrm{x}}$ need not agree, although we prove they are equal in $\mathbb C[x_1, x_2]$.