Tychonoff's theorem

The Tychonoff's theorem (after Andrei Nikolaevich Tikhonov ) is a testimony of the mathematical branch of topology . It reads:

If there is a family of compact topological spaces , then the Cartesian product with the product topology is also compact. ${\displaystyle (X_{i})_{i\in I}}$ ${\displaystyle \textstyle \prod _{i\in I}X_{i}}$

discussion

At first glance, the sentence seems to contradict the opinion. Compactness is in a certain sense a finite property (every open cover has a finite partial cover ), and it may be surprising that this translates into a product with any number of factors. One thinks here of the Riesz's lemma from the functional analysis , after which the closed unit ball of a normed space is compact only in finite spaces, or even the fact that any union of compact sets is generally not compact. What is misleading the view here is the concept of the environment, the "near" in the product topology. Because if a point is near , in the product topology it just means that for a finite number , that is near to . ${\displaystyle \textstyle (x_{i})_{i\in I}\in \prod X_{i}}$ ${\displaystyle \textstyle \left(x_{i}^{(0)}\right)_{i\in I}\in \prod X_{i}}$ ${\displaystyle i_{1},\ldots ,i_{n}}$ ${\displaystyle \in I}$ ${\displaystyle x_{i_{j}}^{(0)}}$ ${\displaystyle x_{i_{j}}}$ ${\displaystyle j=1,\ldots ,n}$

proof

The theorem is particularly easy to prove using ultrafilters : A topological space is compact precisely when every ultrafilter converges on it. Let an ultrafilter be given on the product space. Now look at the image filters under the projections onto the individual rooms. An image filter of an ultrafilter is in turn an ultrafilter, so the sets of points to which the image filters converge are not empty due to the compactness of the individual sets (in the case of Hausdorff areas , the filters have a clear limit value). With the axiom of choice , an element of the product space can then be selected that is the limit value of the respective image filter in each component. This is then also the limit value of the ultrafilter on the product area.

Tychonoff's theorem is also a direct consequence of Alexander's theorem : A space is compact if and only if every cover consisting of elements of a fixed sub-base has a finite partial cover . To show Tychonoff's theorem, one simply considers the sub-base of the sets of elements of the product space that are element of an open set of the respective factor in one component and arbitrary in all other components.

Conversely, it can be shown that the axiom of choice (in ZF ) also follows from Tychonoff's theorem. Note that the theorem for products of compact Hausdorff spaces (which are often just called compact ) does not imply the axiom of choice, because it already follows from the weaker ultrafilter lemma . The above selection is not necessary in this case, as limit values ​​are unambiguous in Hausdorff areas.

Applications

This sentence is used in deriving the following statements:

• Banach-Alaoglu theorem
• Existence of hair measure
• Construction of the Stone-Čech compactification
• From Tychonoff's theorem it follows that the category of compact Hausdorff spaces is complete and the product corresponds to that in the category of topological spaces. These are essential arguments for showing the existence of the Stone-Čech compactification with the theorem about adjoint functors .

literature

• Klaus Jänich : Topology. 4th edition. Springer, Berlin et al. 1994, ISBN 3-540-57471-9 , p. 197 ff.

Web links

• Script on set-theoretical topology (PDF file, German ) (1.72 MB)