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(in mathematics), one of the fundamental concepts of combinatorial topology. It is essential to the aims of this science to regard geometric figures as being subdivided into more elementary figures. It is easiest to construct geometric figures out of simplexes, that is, in the case of three-dimensional space, out of points, lines, triangles, and tetrahedra. Thus, we are most often dealing with simplicial complexes.
A simplicial complex is a finite set of simplexes situated in a certain Euclidean (or Hilbert) space and possessing the following property: the intersection of two simplexes of this set is either empty or is a face of each of them. If a complex contains a γ-dimensional simplex and no simplexes of higher dimension, then the complex is termed γ-dimensional. This very simple concept has undergone many generalizations, proceeding in different directions. Together with the just-defined finite complexes it is possible to define countable complexes. It is further possible to proceed from simplicial complexes to analogously defined cell complexes, whose elements are not necessarily simplexes but any convex polyhedrons or even any figures homeomorphic to them; in the latter case, we speak of “curvilinear” complexes. Ordinarily, only those complexes are considered that satisfy the following closure condition: each face of a simplex belonging to a given complex must also belong to that complex. A set that can be represented as a union of simplexes forming an n-dimensional complex is termed an n-dimensional polyhedron.
REFERENCESAleksandrov, P. S. Kombinatornaia topologiia. Moscow-Leningrad, 1947.
Pontriagin, L. S. Osnovy kombinatornoi topologii. Moscow-Leningrad, 1947.
in psychology, in the most general sense, a particular combination of psychological processes into some sort of whole; in a narrower sense, the word is taken to mean a group of heterogeneous psychic elements connected by a single affect.
Complex, in the latter sense, has become one of the basic concepts of various schools of depth psychology. According to psychoanalysis (S. Freud, Austria), complexes form around tendencies that are displaced to the subconscious (for example, the Oedipus complex arises as a result of the displacement in early childhood of hostile impulses toward the father). Complexes produce deviations in human behavior that are manifested in the form of improper actions, neuroses, and obsessions. In individual psychology (A. Adler, Austria), an exceptional role is attributed to the inferiority complex—an individual’s feeling of his own organic or mental inadequacy. Overcoming this complex by means of compensation is regarded by Adler to be the principal factor in man’s mental development, character formation, and behavior.
D. N. LIALIKOV