By Francesco Costantino

We determine a calculus for branched spines of 3-manifolds through branched Matveev-Piergallini strikes and branched bubble-moves. We in brief point out a few of its attainable purposes within the research and definition of State-Sum Quantum Invariants.

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348]) in 1963; later it was studied by many authors (cf. [27, 28, 36, 55, 75, 115,116, 139, 217, 218, 319, 337]). Below we shall present most general approximation results proved recently in [218] and [217] (see also [36, 76] and [305]). y; v/g; for x; u 2 X and y; v 2 Y: Secondly, we shall use the following result on uniform continuity of maps. 65. Let K be a compact subset of X and let f W X ! Y be a continuous map. K/. Proof. e un are convergent. 66. Let F W X ! Y / be a multi-valued mapping, Z X, and " > 0.

1 we have fxng ! xn ; x0 /: Passing to the limit, as n ! x0 /, as claimed. 79. We recommend [286] and [490] for further details. Next, we present a multi-valued version of the nonlinear alternative of contractive type in Fréchet spaces. n 2 N/. 80. A multi-valued map F W E ! y// Ä kn jx distance. x// C "; for every n 2 N: A subset A E is bounded if for every n 2 N; there exists Mn > 0 such that jxjn Ä Mn ; for every x 2 A. 81. Let E be a Fréchet space, U E an open neighbourhood of the origin, and let N W U !

95. Let r W E ! E/. A/: In other words, we can say that r is a nonexpansive map with respect to the Kuratowski or Hausdorff measure of noncompactness. 96. An / D 0, then A D 1 nD1 An is a nonempty compact subset of E. The following characterisation of Rı -sets, which develops the well-known Hyman theorem [267], was shown by D. Bothe. 97 ([78,397]). Let X be a complete metric space, denotes the measure of noncompactness in X, and let B 6D ;. Bn / ! as n ! 1I (3) B is compact and absolutely neighbourhood contractible.