Graph intersection property
A pair (X,Y) of topological spaces X and Y is said to have the graph intersection property provided that for each continuous function g: XY, if a connected subset of X Y projects onto the whole Y, then it intersect the graph of g. Various relations between this and other known properties related to mapping theory are studied.
In particular, it is proved that: 1) if a space X is completely regular and a space Y is an arcwise connected metric continuum distinct from an arc, then the pair (X,Y) has the graph intersection property if and only if X is hereditary disconnected; 2) if a connected space Y is fixed, then the graph intersection property holds for every pair (X,Y) if and only if there is a closed linear order on Y with minimal and maximal elements. Related results are obtained.
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