Krull topology after Wolfgang Krull. showed that profinite group is isomorphic to one arising from the Galois theory of field but one cannot (yet) control which field will be in this case. In fact, for many fields one does not know in general precisely which finite groups occur as Galois groups over This is the inverse Galois problem for a field (For some fields the inverse Galois problem is settled, such as the field of rational functions in one variable over the complex numbers.) Not every profinite group occurs as an absolute Galois group of a field. The étale fundamental groups considered in algebraic geometry are also profinite groups, roughly speaking because the algebra can only 'see' finite coverings of an algebraic variety. The fundamental groups of algebraic topology, however, are in general not profinite: for any prescribed group, there is a 2-dimensional CW complex whose fundamental group equals it. The automorphism group of a locally finite rooted tree is profinite. Properties and facts Every product of (arbitrarily many) profinite groups is profinite; the topology arising from the profiniteness agrees with the product topology. The inverse limit of an inverse system of profinite groups with continuous transition maps is profinite and the inverse limit functor is exact on the category of profinite groups. Further, being profinite is an extension property. Every closed subgroup of a profinite group is itself profinite; the topology arising from the profiniteness agrees with the subspace topology. If is a closed normal subgroup of a profinite group then the factor group is profinite; the topology arising from the profiniteness agrees with the quotient topology. Since every profinite group is compact Hausdorff, there exists a Haar measure on which allows us to measure the "size" of subsets of compute certain probabilities, and integrate functions on A subgroup of a profinite group is open if and only if it is closed and has finite index. According to a theorem of Nikolay Nikolov and Dan Segal, in any topologically finitely generated profinite group (that is, a profinite group that has a dense finitely generated subgroup) the subgroups of finite index are open. This generalizes an earlier analogous result of Jean-Pierre Serre for topologically finitely generated pro- groups. The proof uses the classification of finite simple groups. As an easy corollary of the Nikolov–Segal result above, surjective discrete group homomorphism between profinite groups and is continuous as long as is