Relatively compact subspaceIn mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) Y of a topological space X is a subset whose closure is compact. Every subset of a compact topological space is relatively compact (since a closed subset of a compact space is compact). And in an arbitrary topological space every subset of a relatively compact set is relatively compact. Every compact subset of a Hausdorff space is relatively compact.
GroupoidIn mathematics, especially in and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: Group with a partial function replacing the binary operation; in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation on the morphisms, called inverse by analogy with group theory. A groupoid where there is only one object is a usual group.
PropertyProperty is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, redefine, rent, mortgage, pawn, sell, exchange, transfer, give away, or destroy it, or to exclude others from doing these things, as well as to perhaps abandon it; whereas regardless of the nature of the property, the owner thereof has the right to properly use it under the granted property rights.
Automorphism groupIn mathematics, the automorphism group of an object X is the group consisting of automorphisms of X under composition of morphisms. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the group of invertible linear transformations from X to itself (the general linear group of X). If instead X is a group, then its automorphism group is the group consisting of all group automorphisms of X. Especially in geometric contexts, an automorphism group is also called a symmetry group.
Equivalence relationIn mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.
Topological groupIn mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other. Topological groups have been studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a very wide class of topological groups.
Locally profinite groupIn mathematics, a locally profinite group is a Hausdorff topological group in which every neighborhood of the identity element contains a compact open subgroup. Equivalently, a locally profinite group is a topological group that is Hausdorff, locally compact, and totally disconnected. Moreover, a locally profinite group is compact if and only if it is profinite; this explains the terminology. Basic examples of locally profinite groups are discrete groups and the p-adic Lie groups.
Covering spaceA covering of a topological space is a continuous map with special properties. Let be a topological space. A covering of is a continuous map such that there exists a discrete space and for every an open neighborhood , such that and is a homeomorphism for every . Often, the notion of a covering is used for the covering space as well as for the map . The open sets are called sheets, which are uniquely determined up to a homeomorphism if is connected. For each the discrete subset is called the fiber of .
Property lawProperty law is the area of law that governs the various forms of ownership in real property (land) and personal property. Property refers to legally protected claims to resources, such as land and personal property, including intellectual property. Property can be exchanged through contract law, and if property is violated, one could sue under tort law to protect it. The concept, idea or philosophy of property underlies all property law.
MonodromyIn mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of monodromy comes from "running round singly". It is closely associated with covering maps and their degeneration into ramification; the aspect giving rise to monodromy phenomena is that certain functions we may wish to define fail to be single-valued as we "run round" a path encircling a singularity.
