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The underlying set of the fundamental group of X is the set of based homotopy classes from the circle to X, denoted [S^1,X]. For general spaces X and Y, there is no natural ...

A general space based on the line element ds=F(x^1,...,x^n;dx^1,...,dx^n), with F(x,y)>0 for y!=0 a function on the tangent bundle T(M), and homogeneous of degree 1 in y. ...

According to many authors (e.g., Kelley 1955, p. 112; Joshi 1983, p. 162; Willard 1970, p. 99) a normal space is a topological space in which for any two disjoint closed sets ...

A topological space that is not connected, i.e., which can be decomposed as the disjoint union of two nonempty open subsets. Equivalently, it can be characterized as a space ...

A curve which may pass through any region of three-dimensional space, as contrasted to a plane curve which must lie in a single plane. Von Staudt (1847) classified space ...

A paracompact space is a T2-space such that every open cover has a locally finite open refinement. Paracompactness is a very common property that topological spaces satisfy. ...

A Banach space X is called minimal if every infinite-dimensional subspace Y of X contains a subspace Z isomorphic to X. An example of a minimal Banach space is the Banach ...

Suppose that X^~,X are arcwise-connected and locally arcwise-connected topological spaces. Then (X^~,p) is said to be a covering space of X if p:X^~->X is a surjective ...

The set of all Schwartz functions is called a Schwartz space and is denoted S(R^n). If C_0^infty(R^n) denotes the set of smooth functions of compact support on R^n, then this ...

Teichmüller's theorem asserts the existence and uniqueness of the extremal quasiconformal map between two compact Riemann surfaces of the same genus modulo an equivalence ...