Unibranch local ring

In algebraic geometry, a local ring "A" is said to be unibranch if the reduced ring "A"red (obtained by quotienting "A" by its nilradical) is an integral domain, and the integral closure "B" of "A"red is also a local ring. A unibranch local ring is said to be geometrically unibranch if the residue field of "B" is a purely inseparable extension of the residue field of "A"red. A complex variety "X" is called topologically unibranch at a point "x" if for all complements "Y" of closed algebraic subsets of "X" there is a fundamental system of neighborhoods (in the classical topology) of "x" whose intersection with "Y" is connected. 
In particular, a normal ring is unibranch. The notions of unibranch and geometrically unibranch points are used in some theorems in algebraic geometry. For example, there is the following result:
Theorem Let "X" and "Y" be two integral locally noetherian schemes and formula_1 a proper dominant morphism. Denote their function fields by "K(X)" and "K(Y)", respectively. Suppose that the algebraic closure of "K(Y)" in "K(X)" has separable degree "n" and that formula_2 is unibranch. Then the fiber formula_3 has at most "n" connected components. In particular, if "f" is birational, then the fibers of unibranch points are connected.
In EGA, the theorem is obtained as a corollary of Zariski's main theorem.