Denotes the one-point compactification

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Reference no: EM132909098

Question: 1 (a) Suppose that X1,......,Xn are topological spaces and A_i, is a subset of X_i, for each i = 1, , n. Prove that int(A₁ x......x int(A_n). Here, int is our notation for taking the interior of a set.

(b) Give an example to show that the equation in part (a) may fail if one considers an infinite product of topological spaces.

Problem 2: Suppose that X and Y are locally compact (but not compact) Hausdorff spaces with one-point compactifications X_∞, and Y_∞ respectively. Further suppose that (X x Y)_∞ denotes the one-point compactification of the locally compact space X x Y.

(a) Let q : X_∞ x Y_∞ → (X x ')_∞ be the map which sends every point of X x Y to itself and sends every other point ∞. Prove that q is a quotient map.

(b) Give an example to how that, in general, X_∞ x Yv is not homeomorphic to (X x Y)_∞.

Reference no: EM132909098

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