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The metric g defined on a nonempty set X by g(x,x) = 0 (1) g(x,y) = 1 (2) if x!=y for all x,y in X. It follows that the open ball of radius r>0 and center at x_0 B(x_0,r)={x ...

The extended complex plane is the name given to the complex plane with a point at infinity attached: C union {infty^~}, where infty^~ denotes complex infinity. It is also ...

A metric on a bunch of segments with a common endpoint O, which defines the distance between two points P_1 and P_2 as the length of the shortest path connecting them inside ...

On the class of topological spaces, a homeomorphism class is an equivalence class under the relation of being homeomorphic. For example, the open interval (-pi/2,pi/2) and ...

The term "homology group" usually means a singular homology group, which is an Abelian group which partially counts the number of holes in a topological space. In particular, ...

Let G denote the group of germs of holomorphic diffeomorphisms of (C,0). Then if |lambda|!=1, then G_lambda is a conjugacy class, i.e., all f in G_lambda are linearizable.

Let K be a finite complex, let h:|K|->|K| be a continuous map. If Lambda(h)!=0, then h has a fixed point.

If F is a sigma-algebra and A is a subset of X, then A is called measurable if A is a member of F. X need not have, a priori, a topological structure. Even if it does, there ...

Somewhere on the Earth, there is a pair of antipodal points having simultaneously the same temperature and pressure.

A distance g on a set that fulfils the same properties as a metric except relaxes the definition to allow the distance between two different points to be zero. An example of ...