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We show that a graph can always be decomposed into edge-disjoint subgraphs of countable car- dinality in which the edge-connectivities and edge-separations of the original graph are preserved up to countable cardinals. We also show that the vertex set of any graph can be endowed with a well-ordering which has a certain compactness property with respect to edge-separation.

The Countable Character of Uncountable Graphs