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By Francesco Costantino

We identify a calculus for branched spines of 3-manifolds via 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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Let f : A + Y be a continuous map, where Y is a regular Hausdorff space. Then f is extendable over X iff the filter base f ( @ ( x ) A A ) converges for each x E X . 9. 11. Theorem. 9. Then a continuous map f : A + Y is extendable over X i y f o r each a E R we have n { c i , f - l ( Y - V ) IV E va}= 0. 6) for all covers Y E Y . 1 1 becomes a theorem proved by Wooten [1973], which he derived from the following theorem. 12. Theorem (Wooten [1973]). Let (Y, Y) be a complete semi-uniform Hausdorff space.

2. C*-, C-, P”- and Pembeddings. . . . . . . . . . . . 3. Unions of C*-embedded subsets . . . . . . . . . . . 4. C*-embedding in product spaces . . . . . . . . . . . 5. Homotopy extension property . . . . . . . . . . . . References.. . . . . . . . . . . . . . . . 41 42 51 62 69 75 78 Introduction In this chapter we shall consider cases, such as C*-, C- or P-embedding, of extending continuous maps defined on a (not necessarily dense) subspace to the whole space.

A) Since we have clearly U{G: 11 E A} c [U{G, I 1 E A}]*, it suffices to prove the converse inclusion. ,*) = X*,a E R and consequently X * - U{Gf 11E A} c St(X - UG,, 4)2:' for a E R. 10 that X* - U(G:IAEA} c Cl(X - U { G , I L E A } ) . 29 we have the desired inclusion [U{Gn I 1 E All* = U { G f I 1 E A}. 4) 23 Extensions of Mappings I (b) Suppose that U{Gt 11 E A} = [U{G, 11 E A}]*. 2) hold in the proof of (a), and for U E 42,, ( X - UG,) n U # 0 iff (X - UG,) n U* # 0, we have U { W * l W E K } = X*.

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