Vol. 41, No. 3, 1972

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ISSN: 0030-8730
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