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Let X be a normed space, M and N be algebraically complemented subspaces of X (i.e., M+N=X and M intersection N={0}), pi:X->X/M be the quotient map, phi:M×N->X be the natural ...

The invariance of domain theorem states that if f:M->N is a one-to-one and continuous map between n-manifolds without boundary, then f is an open map.

If K is a finite complex and h:|K|->|K| is a continuous map, then Lambda(h)=sum(-1)^pTr(h_*,H_p(K)/T_p(K)) is the Lefschetz number of the map h.

Every continuous map f:S^n->R^n must identify a pair of antipodal points.

A nonsingular linear map A:R^n->R^n is orientation-preserving if det(A)>0.

A patch (also called a local surface) is a differentiable mapping x:U->R^n, where U is an open subset of R^2. More generally, if A is any subset of R^2, then a map x:A->R^n ...

A map psi:M->M, where M is a manifold, is a finite-to-one factor of a map Psi:X->X if there exists a continuous surjective map pi:X->M such that psi degreespi=pi degreesPsi ...

A conformal mapping, also called a conformal map, conformal transformation, angle-preserving transformation, or biholomorphic map, is a transformation w=f(z) that preserves ...

Let X and Y be CW-complexes, and let f:X->Y be a continuous map. Then the cellular approximation theorem states that any such f is homotopic to a cellular map. In fact, if ...

A special nonsingular map from one manifold to another such that at every point in the domain of the map, the derivative is an injective linear transformation. This is ...