Algebraic and Geometric Topology, Part 2 by Milgram R. (ed.)

By Milgram R. (ed.)

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5 Minimal and hereditary closure operators Suppose that the M-subobject m : M -. X factors through a larger Msubobject y : Y -+ X as Basic Properties of Closure Operators M my 31 Y X Is it then possible to compute cy(my) with the help of cx(m) ? For the Kuratowski closure operator K of Top this is easily done: for subsets M C Y C X one has ky(M) = Y (1 kx(M) . In the general context, since my = y-' (m) in MfY , one has cy(my) :5 Y-' (cx (m)) by continuity. 4 below). C is called hereditary for X if (HE) cy(my) = y-1(cx(m)) for all m < y in MIX and hereditary if it is hereditary for all X E X .

C is additive 4* C is (l o, 2)-additive, C is fully additive q C is (oo, 2)-additive, C is grounded . C is (1, 0)-additive, 4. for every ec E Card U {oo}, C is (K, ec)-additive, 5. C is (n, A)-additive & A' < A < is < ,' = C is (W, A`)-additive. I. 2. E) and the Lawvere-Tierney topologies or universal closure operations of Topos- and Sheaf Theory (see Chapter 9), as special instances. 5 is not yet apparent in that paper. Earlier papers in Categorical Topology are mostly concerned with particular instances of closure operators, predominantly in the category of topological spaces (see Chapters 6-8).

For integral domains of characteristic p , one has OR = Z . A Prove the formulas (g . f)(m) - g(f(m)) and (g . f)-1(n) = f-1 (g-1 (n)) . Furthermore, for an isomorphism f , verify that f-1 (n) can be interpreted as both, the inverse image of n under f or the image of n under f-1 . $ cf. 6) (a) (Independence of existence of M-pullbacks and of right M-factorizations, Prove that maps in Top may fail to have right M-factorizations for M the class of open embeddings. C (a) (b) Prove that the category CTop does not have M-pullbacks, for M the class of embeddings.

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