Kuratowski set

A one-dimensional set in the plane, which is $ 0 $- 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 $ N ( X) $ of a given $ n $- dimensional space $ X $ consisting of all points $ z \in X $ at which

$$ \mathop{\rm ind} _ {z} X = \mathop{\rm ind} X = n. $$

( $ N( X) $ is called the dimensional kernel of $ X $.) For a metric space $ X $ with a countable base it is always the case that

$$ \mathop{\rm ind} N ( X) \geq \mathop{\rm ind} X - 1, $$

and the Kuratowski set shows that this result is best possible.

The Kuratowski set is constructed as follows. Let $ \Pi $ be the Cantor set in the closed interval $ [ 0, 1] $ of the horizontal axis in a Cartesian coordinate system on the plane. For each

$$ x \in \Pi ,\ \ x = \frac{2}{3 ^ {k _ {1} } } + \frac{2}{3 ^ {k _ {2} } } + \dots ,\ \ k _ {1} < k _ {2} < \dots , $$

put

$$ f ( x) = \ \frac{(- 1) ^ {k _ {1} } }{2} + \frac{(- 1) ^ {k _ {2} } }{2 ^ {2} } + \dots $$

and $ f ( 0) = 0 $. The graph of this function, i.e. the set $ K $ of points $ ( x, f ( x)) $, $ x \in \Pi $, on the plane, is the Kuratowski set. If $ z $ is $ ( x, f( x)) $, where $ x $ is the right-hand end point of an interval adjacent to $ \Pi $, then $ \mathop{\rm ind} _ {z} K = 1 $, but $ \mathop{\rm ind} _ {z} K = 0 $ 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 $ \mathop{\rm ind} N( X) = \mathop{\rm ind} X $ holds) nor connected (it is even totally disconnected, since it admits a continuous one-to-one mapping onto $ \Pi $: $ \langle x, f( x)\rangle \rightarrow x $).

The Kuratowski set is, however, completely metrizable.

A separable metric space $ X $ such that $ \mathop{\rm ind} X = n \geq 1 $ and such that the dimensional kernel of $ X $ has dimension $ n- 1 $, is sometimes called weakly $ n $- dimensional.

The Kuratowski set is weakly $ 1 $- dimensional.

The first example of such a space was given by W. Sierpiński [a2]. For weakly $ n $- dimensional spaces for $ n = 2 , 3 \dots $ 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 -dimensional spaces" Fund. Math. , 103 (1979) pp. 1–8