The well-known Jordan Curve Theorem says that a simple closed curve in the
plane cuts the plane into two regions and is their common boundary. Thus, it
is a cutting of the plane. The so called Warsaw Circle, obtained from the
$\sin (1/x)$-curve
$$S = \{(0,y) \in \Bbb R^2: y \in [-1,1]\} \cup \{(x,\sin(1/x))
\in \Bbb R^2: x \in (0,1] \}$$
by identifying the points $(0,-1)$ and $(1, \sin 1)$ also is
a cutting. But both these continua contain arcs, which
certainly do not cut the plane. So, one can ask a question
if there exists such a cutting $X \subset \Bbb R^2$ that
each nondegenerate subcontinuum of $X$ also cuts $\Bbb R^2$.
The question has been answered in the affirmative in 1930 by
G. T. Whyburn who constructed in [3] a {\it hereditary
cutting of the plane}, which is now called the \h{Whyburn's
Curve}. The present description of this peculiar continuum
is a modification of the original one due to A. Lelek [1],
to obtain some extra properties of the curve, and it follows
Lelek's sketch of the construction in [3, p. 117-119]. We
say that a continuum $X$ lying in the plane $\Bbb R^2$ cuts
the plane between points $a$ and $b$ if the points belong to
different components of $\Bbb R^2 \setminus X$. A continuum
$X$ is a cutting of $\Bbb R^2$ if there are some two points
such that $X$ cuts $\Bbb R^2$ between them. Take the
$\sin(1/x)$curve $S$ and identify points $(0,1)$ and
$(0,-1)$ into one point $r$. The obtained curve (still
considered as a subset of the plane) consists of a circle
$C$ and a sinusoidal curve approximating the circle, so it
is a cutting of the plane. If we add to the curve an arc
starting at $r$ and having only this point in common with
the curve, we get a continuum $Y$ being also a cutting of
the plane (namely it cuts the plane between a point $a$
inside the circle $C$ and a point $b$ lying in $\Bbb R^2
\setminus Y$ outside $C$) and having the property that every
subcontinuum of $Y$ containing a point $p$ of the sinusoidal
part of $Y$ and a point $q$ lying on the added arc, also
cuts the plane between $a$ and $b$ (see Figure A (I)-(II).).
\par The same curve $Y$ can also be obtained as the
intersection of a nested sequence of plane continua. Each
continuum of the sequence has the form of a long strip with
one loop and finitely many zig-zags. Consecutive strips are
thinner and thinner, more and more narrow, each next being
inscribed in the previous one in such a way that the next
loop is contained in the previous loop which also contains
one more zig-zag of the next strip (see Figure A
(III)-(IV)). Consequently, the number of zig-zags in the
next approximation is greater than that in the previous one,
so these numbers tend to infinity. The common part of the
all strips is just the curve $Y$, while the common part of
the loops is the circle $C$ contained in $Y$. \par

The Whyburn's curve $W$ can be obtained in a similar way, as
the intersection of a nested sequence of strips in the
plane, using the method called "condensation of
singularities". Namely the singularity appearing in each
neighborhood of $C$ in $Y$ is copied more and more times
both in the loops as well as in the zig-zag parts of the
consecutive strips. A little bit more precisely speaking,
the strips are modified so that each next strip, magnifying
the number of zig-zags in the loops and preserving previous
loops, contains also new, smaller loops, distributed in a
more and more dense manner inside the previous strip (see
Figure A (V)) so that the diameters of the loops tend to
zero as the number of steps of the construction tends to
infinity. As a consequence of this property, for every two
points $p$ and $q$ of the curve $W$ one can find, in a
sufficiently far approximating strip, a loop lying between
these points. Roughly speaking, behavior of the curve $W$
between $p$ and $q$ is such as that of the curve $Y$, i.e.,
each subcontinuum of $W$ containing $p$ and $q$ cuts the
plane between some points $a$ and $b$ determined by the
mentioned loop. \par The continuum $W$ constructed is this
way has, among others, the following properties (see [1]).
1) each subcontinuum of $W$ is a cutting of the plane, and
therefore $W$ contains no arc; 2) each subcontinuum of $W$
contains a homeomorphic copy of $W$; 3) $W$ is hereditarily
decomposable; and 4) $W$ is a continuous image of the
pseudo-arc $\Bbb P$. Since neither the pseudo-arc $\Bbb P$
nor any subcontinuum of $\Bbb P$ cuts the plane, this last
property shows that a hereditary anti-cutting of $\Bbb R^2$
can be continuously mapped onto a hereditary cutting.

\medskip


[1]
 A. Lelek :  On weakly chainable continua, Fund. Math. 51
 (1962), 271--282.


[2] A. Lelek : Zbiory, PZWS, Warszawa 1966.

[3] G. T. Whyburn : A continuum every subcontinuum of which
separates the plane, Amer. J. Math. 52 (1930), 319--330.
