Katětov's interpolation theorem

The interpolation set of Katětov ( English : Katětov's interpolation theorem ) is a theorem , which is the mathematical sub-area of topology attributable. It goes back to the Czech mathematician Miroslav Katětov and gives a generalization of Tietze's well-known continuation theorem .

Formulation of the sentence

The sentence can be formulated as follows:

Let a normal topological space be given . ${\displaystyle X}$

Let two semi-continuous real-valued functions be given and it is assumed that it is above- semi-continuous , that it is below- semi-continuous and that the inequality always exists. ${\displaystyle g,h\colon X\to \mathbb {R} }$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle g(x)\leq h(x)\;\;(x\in X)}$

Then there is a continuous function which interpolates and , ${\displaystyle f\colon X\to \mathbb {R} }$ ${\displaystyle g}$ ${\displaystyle h}$

for which so pointwise the inequality

${\displaystyle g\leq f\leq h}$

consists.

Remarks

• Katětov's interpolation theorem leads to the continuation of Tietzes as a consequence. For this purpose one shows the continuation theorem with the help of the interpolation theorem for continuous functions which map a closed subset of a normal topological space into the interval . Then - with known methods - one obtains the continuation theorem for all continuous functions that map a closed subset of a normal topological space according to or (more generally) into a product space consisting of real intervals . ${\displaystyle [0,1]}$ ${\displaystyle \mathbb {R} }$
• In his work from 1951, Katětov made a mistake in deriving his interpolation theorem, which was corrected by Hing Tong in his work from 1952. In the English-language literature, the interpolation theorem is therefore often assigned to both authors mentioned and then - by Tomasz Kubiak (see below) - referred to as Katětov-Tong insertion theorem .
• The interpolation theorem can also be derived with the help of Urysohn's lemma . Since Urysohn's lemma and Tietze's continuation theorem are essentially equivalent and since the interpolation theorem entails the continuation clause - as we have seen - all three theorems thus even turn out to be equivalent. As can be seen (not least on the basis of the works mentioned), these essentially mean that a normal room always has the functional properties claimed in the three theorems and that these characterize it.

literature

• Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).
• GJO Jameson : Topology and Normed Spaces . Chapman and Hall, London 1974, ISBN 0-412-12880-2 ( MR0463890 ).
• M. Katětov: On real-valued functions in topological spaces . In: Fund. Math . tape 38 , 1951, pp. 85-91 ( MR0050264 ).
• Tomasz Kubiak : A stiffening of the Katětov-Tong insertion theorem . In: Comment. Math. Univ. Carolin . tape 34 , 1993, pp. 357-362 ( [1] ). MR1241744
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).
• Hing Tong: Some characterizations of normal and perfectly normal spaces . In: Duke Math J. . tape 19 , 1952, pp. 289-292 ( MR0050265 ).

Individual evidence

1. ↑ ^{a b c} G. JO Jameson: Topology and Normed Spaces. 1974, pp. 120-123.
2. ↑ So is a topological space in which any two disjoint closed sets by disjoint open neighborhoods are separated . ${\displaystyle X}$
3. ↑ With regard to the latter assumption, one says that the inequality exists point-wise or element-wise .
4. ↑ Jameson, op. Cit., Pp. 113-115, 123.
5. ↑ Horst Schubert: Topology. 1975, p. 78 ff
6. ↑ This general version of the continuation clause is also called the Tietze-Urysohn theorem .
7. ↑ Horst Schubert: Topology. 1975, pp. 76-83