Notes on general topology

Extract

It is well known that a path-connected Hausdorff space is arc-connected. Indeed, given a path P, that is, a continuous map. f of the unit interval I₀(0 ≤ t ≤ 1) into the Hausdorff space X, there is an arc joining the points f(0), f(1) (supposed distinct) which is obtained by ‘cutting out loops in P’. More precisely, there exist a continuous increasing map α of I₀ onto itself and a homeomorphism φ of I₀ onto φ(I₀) ⊂ X, such that if [t₀, t₁] is any maximal interval of constancy of α (including the case t₀ = t₁) then f(t₀) = f(t₁) = φοα(t₀). The function φ, α, can be defined by a constructive process.