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Notes on general topology

II. A generalization of arc-connectedness

Published online by Cambridge University Press:  24 October 2008

A. J. Ward
Affiliation:
Emmanuel College, Cambridge

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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1965

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