By Francesco Costantino

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

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Topology and Limits As we all know, the definition of limit used in real analysis reads as follows: The sequence of numbers (av)veN converges to the limit p if and only if, given an arbitrary e > 0, we can find an integer n such that v ^ n implies \av — p | < e. The set U((p) = {x : \ x — p | < c} is called an e-neighborhood of p. The convergence of the sequence {av}V£N to the limit p is equivalent to the following: Given any € > 0 there exists a terminating section An of the given sequence such that An ç: U€(p).

50 / / . Topo/ogica/ Spaces to what follows). Let N be the set of all x G M such that p ^ x < q and such that if p ^ y ^ x, then y e M. Since p e N} N Φ &. And since q is an upper bound of TV, TV possesses a sup s. Every neighborhood belonging to rs contains a point of N and also a point of UM. Hence s e N c M and 5 G UM. If M were both open and closed, we would have s e Mn \JM = M n u M , a contradiction. (2)—>-(l): Suppose a < b. If there exists no p e R such that a

10 Proof: (1)—>(2): Let M be a set different from 0 and from R. (if<7 < /), the proof would be analogous Condition (1) will serve later to characterize the notion of connectedness (Section 14). 50 / / . Topo/ogica/ Spaces to what follows). Let N be the set of all x G M such that p ^ x < q and such that if p ^ y ^ x, then y e M. Since p e N} N Φ &. And since q is an upper bound of TV, TV possesses a sup s. Every neighborhood belonging to rs contains a point of N and also a point of UM. Hence s e N c M and 5 G UM.