by identifying the points (0,-1) and 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 that each nondegenerate subcontinuum of

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*.

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.

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 . Since neither the pseudo-arc
nor any subcontinuum of cuts the plane, this last
property shows that a hereditary anti-cutting of
can be continuously mapped onto a hereditary cutting.

[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.