Tychonoff's theorem

The set of Tychonoff (after Andrei Nikolaevich Tikhonov ) is a statement from the mathematical branch of topology. It reads:

Is a family of compact topological spaces, then the Cartesian product with the product topology is compact.

Discussion

The sentence seems to contradict at first glance of intuition. Compactness is in some ways a finiteness property ( every open cover has a finite subcover ), and it may seem surprising that this translates to a product with any number of factors. One thinks here of the lemma of Riesz from functional analysis, after which the closed unit ball of a normed space is compact only in finite dimensional spaces, or even the fact that an arbitrary union of compact sets in general is no longer compact. What the intuition here leads astray, is the notion of environment, the " near " in the product topology. Since if a point is in the vicinity of, that is, in the product topology just only applies to many finite that is near to.

Evidence

The set is particularly easy to prove by Ultra Filter: A topological space is compact if every ultrafilter converges on it. Be given an ultrafilter on the product space. Consider now the image filter under the projections onto the individual rooms. An image filter of an ultrafilter is an ultrafilter in turn, are therefore the quantities of the items against which the image filters converge due to the compactness of the individual amounts not empty (in the case of Hausdorff spaces have the filter has a unique limit ). With the axiom of choice, it is then an element of the product space filter, which is limit of the respective image filter in each component. This is then also limit the ultrafilter on the product space.

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

Conversely, it can be shown that the axiom of choice ( in ZF ) follows from the theorem of Tychonoff. Note that, however, the set of products of compact Hausdorff spaces (often also just called compact) does not imply the axiom of choice, because it follows from the weaker Ultrafilterlemma. The above selection is not necessary in this case because limits are clearly in Hausdorff spaces.

Applications

This phrase is used in the derivation of the following statements:

• Banach - Alaoglu
• Existence of the hair dimension
• Construction of the Stone - Čech compactification
• It follows from the theorem of Tychonoff, that the category of compact Hausdorff spaces is complete and the product is in conformity with the in the category of topological spaces. These are the main arguments in order to show the theorem on adjoint functors the existence of Stone - Čech compactification can.