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Locally Sierpinski quotients

Given any non-trivial, connected topological space X, it is possible to de ne an equivalence relation ~ on it such that the topological quotient space X/ ~ is the Sierpinski space. Locally Sierpinski spaces are generalizations of the Sierpinski space and here we address the following question. Does a statement like the one above hold if Sierpinski is replaced by (proper) locally Sierpinski? The answer is no and...