Compactly generated shape

Compactly Generated Shape Index Theory and its Application to a Retarded Nonautonomous Parabolic Equation

“…For the presentation hereafter, we first introduce the definition of H-shape of pairs of Hausdorff spaces here, see [4] for the case of Hausdorff spaces and originally [19].…”

“…We refer to this definition of shape given by Mardešić and Segal as ANR-shape, denoted by Sh ANR . In 1974 Rubin and Sanders gave a different extension to the realm of Hausdorff spaces, called "compactly generated shape", shortly H-shape, see [19]. The establishment of H-shape theory is based on the ANR-shape theory of compact Hausdorff spaces.…”

“…In the following, we introduce the definition of H-shape for pairs of Hausdorff spaces in detail as an extension of H-shape for Hausdorff spaces ( [4,19]).…”

“…Given two compact pairs (K, K 0 ) and (L, L 0 ), there is a shape map f : (K, K 0 ) → (L, L 0 ), which is defined as follows, see [4,15,19]. The shape map f assigns to every pair (Q, Q 0 ) having the homotopy type of a CW-complex pair and to every homotopy…”

“…It is clear that the shape index pair can be constructed by using local unstable manifolds W u N (K) and their appropriate sections, since the compactness of W u N (K) ( [20]) can make the quotient space metrizable. In this paper, we employ the compactly generated shape defined for general Hausdorff spaces in Rubin and Sanders [19], which allows us to remove the additional assumptions in [27], such as the separability of the phase space and the compactness of the exit set (this was not mentioned clearly in [27]). Then we develop a new type of shape index theory for local semiflows on complete metric spaces via much more general index pairs (N, E).…”