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We construct an epsilon coincidence theory which generalizes, in some aspect, the epsilon fixed point theory proposed by Robert Brown in 2006. Given two maps f, g: X → Y from a well-behaved topological space into a metric space, we define µ ∈(f, g) to be the minimum number of coincidence points of any maps f 1 and g 1 such that f 1 is ∈ 1-homotopic to f, g 1 is ∈ 2-homotopic to g and ∈ 1 + ∈ 2 < ∈. We prove that if Y is a closed Riemannian manifold, then it is possible to attain µ ∈(f, g) moving...

Given a model 2-complex K P of a group presentation P, we associate to it an integer matrix ΔP and we prove that a cellular map f: K P → S 2 is root free (is not strongly surjective) if and only if the diophantine linear system ΔP Y = $$\overrightarrow{deg}$$ (f) has an integer solution, here $$\overrightarrow{deg}$$ (f)is the so-called vector-degree of f

For a multivalued map $\varphi :Y⊸(X,\tau )$ between topological spaces, the upper semifinite topology $𝒜\left(\tau \right)$ on the power set $𝒜\left(X\right)=\{A\subset X:A\ne \varnothing \}$ is such that $\varphi$ is upper semicontinuous if and only if it is continuous when viewed as a singlevalued map $\varphi :Y\to \left(𝒜\right(X),𝒜(\tau \left)\right)$. In this paper, we seek a result like this from a reverse viewpoint, namely, given a set $X$ and a topology $\Gamma$ on $𝒜\left(X\right)$, we consider a natural topology $ℛ\left(\Gamma \right)$ on $X$, constructed from $\Gamma$ satisfying $ℛ\left(\Gamma \right)=\tau$ if $\Gamma =𝒜\left(\tau \right)$, and we give necessary and sufficient conditions to the upper semicontinuity of a multivalued map $\varphi :Y⊸(X,ℛ(\Gamma \left)\right)$...