### ${\mathcal{C}}_{p}-E$-movable and $\mathcal{C}-E$-calm compacta and their images

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We consider alternating sums of squares of odd and even terms of the Lucas sequence and alternating sums of their products. These alternating sums have nice representations as products of appropriate Fibonacci and Lucas numbers.

We shall describe a modification of homotopy theory of maps which we call shape theory of maps. This is accomplished by constructing the shape category of maps HMb. The category HMb is built using multi-valued functions. Its objects are maps of topological spaces while its morphisms are homotopy classes of collections of pairs of multi-valued functions which we call multi-binets. Various authors have previously given other descriptions of shape categories of maps. Our description is intrinsic in...

In this paper we shall introduce notions of F-universality and F-e-universality for maps between compact Hausdorff spaces and explore the behaviour of these properties under the operation of composition of maps. We consider both the quest for conditions on maps f and g which would imply that their composition g o f is either F-universal or F-e-universal and the quest for consequences on f and g when the composition g o f is either F-universal or F-e-universal. In our approach F is an arbitrary class...

We prove in this paper that the category HM whose objects are topological spaces and whose morphisms are homotopy classes of multi-nets is naturally equivalent to the shape theory Sh. The description of the category HM was given earlier in the article "Shape via multi-nets". We have shown there that HM is naturally equivalent to Sh only on a rather restricted class of spaces. This class includes all compact metric spaces where a similar intrinsic description of the shape category using multi-valued...

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