On Vietoris mapping theorem and its inverse

A Vietoris Mapping Theorem for Homotopy

Let X and Y be compact metric spaces and let a map /: X—> Y be onto. The Vietoris Mapping Theorem as proved by Vietoris [8] states that if for all Ofkrfkn-1 and all yEY, flr(/_1(y)) =0 (augmented Vietoris homology mod two) then the induced homomorphism /*: Hr(X)-+Hr(Y) is an isomorphism onto for rfkn — l and onto for r=n. Begle [l; 2] has generalized this theorem to nonmetric spaces and more ge...

A note on spectral mapping theorem

This paper aims to present the well-known spectral mapping theorem for multi-variable functions.

Determinants, Matrix Norms, Inverse Mapping Theorem

The purpose of this notes is to present some useful facts about matrices and determinants and a proof of the inverse mapping theorem that is rather different from the one in Apostol. Notation: Mn(R) denotes the set of all n× n real matrices. Determinants: If A ∈ Mn(R), we can consider the rows of A: r1, . . . , rn. These are elements of R, considered as row vectors. Conversely, given n row vect...

An Inverse Mapping Theorem for Set-valued Maps

We prove that certain Lipschitz properties of the inverse F-1 of a set-valued map F are inherited by the map (f+F)~x when / has vanishing strict derivative. In this paper, we present an inverse mapping theorem for set-valued maps F acting from a complete metric space I toa linear space Y with a (translation) invariant metric. We prove that, for any function f: X -> Y with "vanishing strict deri...

On the Inverse Function Theorem