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A subset S of a topological space X is compact if for every open cover of S there exists a finite subcover of S.

Two geometric figures are said to be concentric if their centers coincide. The region between two concentric circles is called an annulus. The following table summarizes some ...

Given a triangle, draw a Cevian to one of the bases that divides it into two triangles having congruent incircles. The positions and sizes of these two circumcircles can then ...

A topological space decomposes into its connected components. The connectedness relation between two pairs of points satisfies transitivity, i.e., if a∼b and b∼c then a∼c. ...

A connected set is a set that cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set. Equivalently, it is a set which ...

A space D is connected if any two points in D can be connected by a curve lying wholly within D. A space is 0-connected (a.k.a. pathwise-connected) if every map from a ...

A set in Euclidean space R^d is convex set if it contains all the line segments connecting any pair of its points. If the set does not contain all the line segments, it is ...

A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the ...

Geometric objects lying in a common plane are said to be coplanar. Three noncollinear points determine a plane and so are trivially coplanar. Four points are coplanar iff the ...

The complex lattice Lambda_6^omega corresponding to real lattice K_(12) having the densest hypersphere packing (kissing number) in twelve dimensions. The associated ...