Kuratowski set
A one-dimensional set in the plane, which is -dimensional at all its points with the exception of a countable set. First constructed by C. Kuratowski [1] in connection with the problem of the dimension of the subset
of a given
-dimensional space
consisting of all points
at which
![]() |
( is called the dimensional kernel of
.) For a metric space
with a countable base it is always the case that
![]() |
and the Kuratowski set shows that this result is best possible.
The Kuratowski set is constructed as follows. Let be the Cantor set in the closed interval
of the horizontal axis in a Cartesian coordinate system on the plane. For each
![]() |
put
![]() |
and . The graph of this function, i.e. the set
of points
,
, on the plane, is the Kuratowski set. If
is
, where
is the right-hand end point of an interval adjacent to
, then
, but
at all other points.
References
[1] | C. Kuratowski, "Une application des images de fonctions à la construction de certains ensembles singuliers" Mathematica , 6 (1932) pp. 120–123 |
[2] | P.S. Aleksandrov, B.A. Pasynkov, "Introduction to dimension theory" , Moscow (1973) (In Russian) |
Comments
The Kuratowski set is not a continuum, since it is neither compact (for compact metric spaces the equality holds) nor connected (it is even totally disconnected, since it admits a continuous one-to-one mapping onto
:
).
The Kuratowski set is, however, completely metrizable.
A separable metric space such that
and such that the dimensional kernel of
has dimension
, is sometimes called weakly
-dimensional.
The Kuratowski set is weakly -dimensional.
The first example of such a space was given by W. Sierpiński [a2]. For weakly -dimensional spaces for
cf. [a3], [a4].
References
[a1] | R. Engelking, "Dimension theory" , North-Holland & PWN (1978) pp. 19; 50 |
[a2] | W. Sierpiński, "Sur les ensembles connexes et non-connexes" Fund. Math. , 2 (1921) pp. 81–95 |
[a3] | S. Mazurkiewicz, "Sur les ensembles de dimension faibles" Fund. Math. , 13 (1929) pp. 210–217 |
[a4] | B. Tomaszewski, "On weakly ![]() |
Kuratowski set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kuratowski_set&oldid=12528