This kind of curiosities in continuum theory is connected
with common boundary of several plane regions. In 1904 A.
Schönflies started to publish a sequence of papers
[6] which became an important step in development of
continuum theory. Relying heavily on intuition, he claimed
that there do not exist three regions in the plane with
common boundary. The claim was refused by L. E. J. Brouwer
in 1910 [1] who constructed continua which are common
boundary of three regions and showed that they are *
indecomposable*, i.e., such continua *X* that there are no
nonempty proper subcontinua *A* and *B* of *X* with . The first example of such a continuum was given by
Brouwer in 1910 (see Figure B, where the first several steps
of his construction, simplified by Z. Janiszewski
(1888-1920) [2, p. 114] are presented). Finally in
1922 B. Knaster (1893-1980) gave a nice description of this
continuum (with the first full proof of its
indecomposability) in [3, p. 209-210]. Now the
continuum is referred to as the simplest indecomposable
continuum, or the horse-shoe continuum, or the B-J-K
continuum (for Brouwer, Janiszewski and Knaster).

Finally the common boundary problem of plane domains has
been solved in 1928 by K. Kuratowski (1896-1980) who proved
in [4] and [5] that every plane continuum
which is the common boundary of *n* open domains either is
indecomposable or is the union of two indecomposable
continua whenever , and when *n* = 2 it either is "monostratic" or has a natural "cyclic
structure" in the sense that it is built up from layers naturally ordered in
the same way as the individual points of the circle.

We can obtain a homeomorphic copy of the buckethandle continuum using the inverse limit

where for each let