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To see the sufficiency, suppose that every net in X has a converging subnet. Then for arbitrary family {Fi |i ∈ I} of closed sets in X having the finite intersection property, if we consider a net on the set of finite subsets of I directed by inclusion as S : F (I) A → S(A) ∈ i∈A Fi , the limit point of a converging subnet of S is easily seen to belong to all Fi , i ∈ I. In Euclidean n-space, a closed bounded set is compact. 13 So it is useful to remember the next property on the homeomorphism between compact spaces.

Assume that for each x ∈ / Fix(ϕ), convex set Ψ(x) exists such that ϕ(x) ⊂ Ψ(x) and x ∈ / Ψ(x). Then ϕ has a fixed point. 5, convexity for the value of ϕ in Browder’s theorem is replaced by the existence of convex set Ψ(x) including ϕ(x) at each x such that x ∈ X\Fix(ϕ). Note also that Ψ may be considered a correspondence satisfying a local intersection property on X\Fix(ϕ). The key concept behind the theorem is a characterization of mappings relative to the set of their fixed points. 5, therefore, it is sufficient to show the next theorem, which is written through further general concepts and properties for correspondence ϕ relative to the set of its fixed points.

Let βt : X → [0, 1], t = 1, . . , n be a partition of unity subordinate to {U 1 , . . , U n }. For each z ∈ X, define Ψ(z) as n Ψ(z) = y∈X βt (z)pt (y − z) > 0 . t=1 For all z ∈ X, for all y ∈ ϕ(z), and for all t such that z ∈ Ut , we have pt (y − n n z) > 0, so that t=1 βt (z)(pt (y − z)) = [ t=1 βt (z)pt ](y − z) > 0. That is, Ψ(z) ⊃ ϕ(z) for all z ∈ X. Therefore, Ψ : X → X is a fixed-point-free convex extension of ϕ on X. Moreover, Ψ has a local intersection property on X\Fix(ϕ) = X, since if x ∈ X and y x ∈ ϕ(x), open neighborhood V (x) of x in X exists such that ∀z ∈ V (x), y x ∈ Ψ(z).