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A quotient map which is neither open nor closed The example in Exercise 3 on page 145 of Munkres is given by taking A = R × {0} ∪ [0, ∞] × R and letting f : A → R send (x, y) ∈ A to x. It is not closed, for if B is the hyperbola defined by the equation xy = 1 and C = A ∩ B then C is closed in A but its image under f is the nonclosed set (0, ∞). It is not open, for if W ⊂ R2 is the open rectangular region (−2, 2)×(1, 2), then V = W ∩A is open in A but its image under f is the nonopen subset [0, 2). There are several ways to check that f is a quotient map. The quickest is to use the preceding exercise; by the latter, if we can find a map σ : R → A such that f o σ is the identity, then f is a quotient map. If we take σ(x) = (x, 0), then σ has the required property.