Let
R
R
be a reduced affine
C
\mathbb {C}
-algebra with corresponding affine algebraic set
X
X
. Let
C
(
X
)
\mathcal {C}(X)
be the ring of continuous (Euclidean topology)
C
\mathbb {C}
-valued functions on
X
X
. Brenner defined the continuous closure
I
c
o
n
t
I^{\mathrm {cont}}
of an ideal
I
I
as
I
C
(
X
)
∩
R
I\mathcal {C}(X) \cap R
. He also introduced an algebraic notion of axes closure
I
a
x
I^{\mathrm {ax}}
that always contains
I
c
o
n
t
I^{\mathrm {cont}}
, and asked whether they coincide. We extend the notion of axes closure to general Noetherian rings, defining
f
∈
I
a
x
f \in I^{\mathrm {ax}}
if its image is in
I
S
IS
for every homomorphism
R
→
S
R \to S
, where
S
S
is a one-dimensional complete seminormal local ring. We also introduce the natural closure
I
♮
I^{\natural }
of
I
I
. One of many characterizations is
I
♮
=
I
+
{
f
∈
R
:
∃
n
>
0
w
i
t
h
f
n
∈
I
n
+
1
}
I^{\natural } = I + \{f \in R: \exists n >0 \mathrm {\ with\ } f^n \in I^{n+1}\}
. We show that
I
♮
⊆
I
a
x
I^{\natural } \subseteq I^{\mathrm {ax}}
and that when continuous closure is defined,
I
♮
⊆
I
c
o
n
t
⊆
I
a
x
I^{\natural } \subseteq I^{\mathrm {cont}} \subseteq I^{\mathrm {ax}}
. Under mild hypotheses on the ring, we show that
I
♮
=
I
a
x
I^{\natural } = I^{\mathrm {ax}}
when
I
I
is primary to a maximal ideal and that if
I
I
has no embedded primes, then
I
=
I
♮
I = I^{\natural }
if and only if
I
=
I
a
x
I = I^{\mathrm {ax}}
, so that
I
c
o
n
t
I^{\mathrm {cont}}
agrees as well. We deduce that in the polynomial ring
C
[
x
1
,
…
,
x
n
]
\mathbb {C} \lbrack x_1, \ldots , x_n \rbrack
, if
f
=
0
f = 0
at all points where all of the
∂
f
∂
x
i
{\partial f \over \partial x_i}
are 0, then
f
∈
(
∂
f
∂
x
1
,
…
,
∂
f
∂
x
n
)
R
f \in ( {\partial f \over \partial x_1}, \, \ldots , \, {\partial f \over \partial x_n})R
. We characterize
I
c
o
n
t
I^{\mathrm {cont}}
for monomial ideals in polynomial rings over
C
\mathbb {C}
, but we show that the inequalities
I
♮
⊆
I
c
o
n
t
I^{\natural } \subseteq I^{\mathrm {cont}}
and
I
c
o
n
t
⊆
I
a
x
I^{\mathrm {cont}} \subseteq I^{\mathrm {ax}}
can be strict for monomial ideals even in dimension 3. Thus,
I
c
o
n
t
I^{\mathrm {cont}}
and
I
a
x
I^{\mathrm {ax}}
need not agree, although we prove they are equal in
C
[
x
1
,
x
2
]
\mathbb {C}[x_1, x_2]
.