Topological Continuous Functions - Mathematics Stack Exchange The pasting lemma for finitely many closed sets now says that h h is continuous on X X (a) would follow from the following lemma: If Y Y is an ordered topological space, L = {(y,y′) ∈Y2: y ≤y′} L = {(y, y) ∈ Y 2: y ≤ y} is closed in Y2 Y 2 Assuming this lemma, (a) follows from standard facts on the product topology: