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The opening of the dark-blue square by a disk, resulting in the light-blue square with round corners. In mathematical morphology, opening is the dilation of the erosion of a set A by a structuring element B:
Mathematical Morphology was developed in 1964 by the collaborative work of Georges Matheron and Jean Serra, at the École des Mines de Paris, France.Matheron supervised the PhD thesis of Serra, devoted to the quantification of mineral characteristics from thin cross sections, and this work resulted in a novel practical approach, as well as theoretical advancements in integral geometry and ...
The open, closed, regular open, regular closed and clopen elements of the interior algebra A(X) are just the open, closed, regular open, regular closed and clopen subsets of X respectively in the usual topological sense. Every complete atomic interior algebra is isomorphic to an interior algebra of the form A(X) for some topological space X.
A morphism f:X→Y of algebraic varieties is said to be dominant if it has dense image. For such an f, if V is a nonempty open affine subset of Y, then there is a nonempty open affine subset U of X such that f(U) ⊂ V and then #: [] [] is injective.
However, if f is faithfully flat and quasi-compact, it is not in general true that f is open, even if X and Y are noetherian. [27] Furthermore, no converse to this statement holds: If f is the canonical map from the reduced scheme X red to X , then f is a universal homeomorphism, but for X non-reduced and noetherian, f is never flat.
In mathematics, many types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a ...
This follows from the going up theorem of Cohen-Seidenberg in commutative algebra. Finite morphisms have finite fibers (that is, they are quasi-finite). [4] This follows from the fact that for a field k, every finite k-algebra is an Artinian ring. A related statement is that for a finite surjective morphism f: X → Y, X and Y have the same ...
A good cover is an open cover in which all sets and all non-empty intersections of finitely-many sets are contractible (Petersen 2006). The concept was introduced by André Weil in 1952 for differentiable manifolds , demanding the U α 1 … α n {\displaystyle U_{\alpha _{1}\ldots \alpha _{n}}} to be differentiably contractible.