Vol. 41, No. 3, 1972

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An adjunction theorem for locally equiconnected spaces

Eldon Dyer and S. Eilenberg

Vol. 41 (1972), No. 3, 669–685

The locally equiconnected spaces (LEC spaces) can be characterized as the spaces X with the property that if f0,f1 : Z X are mappings which are “sufficiently close together” and which agree on a subspace A of Z, then f0 is homotopic to f1 relative to A(f0f1 rel A); i.e., there is a morphism F : Z × I X with F|Z × 0 = f0,F|Z × 1 = f1 and F(a,t) = f0a for all a A and t I.

The notion of “close” is measured by a morphism φ : X ×X I with φ(x,x) = 0 if and only if x = x. We then require that φ(f0ξ,f1ξ) < 1 for all ξ Z implies that f0f1 rel A.

There is a universal test pair (u,D); let u = φ1[0,1) and D = φ10. Let f0 and f1 be the restrictions to u of the projections X × X X onto the first and second coordinates. Then f0 and f1 agree precisely on D. A homotopy f0f1 relD exists if and only if D is a strong deformation retract of u in X × X.

We note two things. First, the existence of a homotopy f0f1 rel D implies the existence of the homotopies in the general case described in the first paragraph. Second, the existence of a map φ and homotopy f0f1 rel D is equivalent to the diagonal map

Δ : X → X × X

being a cofbration.

Mathematical Subject Classification 2000
Primary: 54D05
Secondary: 55F05
Received: 5 February 1971
Revised: 19 May 1971
Published: 1 June 1972
Eldon Dyer
S. Eilenberg