Splittability for Partially Ordered Sets
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A topological space X is said to be splittable over a class P of spaces if for every A⊂X there exists continuous f:X→Y∈P such that f(A)∩f(X∖A) is empty. A class P of topological spaces is said to be a splittability class if the spaces splittable over P are precisely the members of P. We extend the notion of splittability to partially ordered sets and consider splittability over some elementary posets. We identify precisely which subsets of a poset can be split along over an n-point chain. Using these results it is shown that the union of two splittability classes need not be a splittability class and a necessary condition for P to be a splittability class is given.
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