Q1. Let X be Hausdorff and Y = X ∪ {∞} be its one-point compactification.
(a) Show that if X is not compact, then X- = Y.
(b) Show that if X is compact, then X- = X, and Y is disconnected with {∞} being one of its components.
Q2. Prove or disprove:
(a) Q ∪ {∞} is Hausdorff.
(b) Z with the digital line topology is compact.
(c) Z with the arithmetic topology is compact.
(d) (0, 1) ∪ {∞} is homeomorphic to the circle.
(e) (0.1] ∪ {∞} is homeomorphic to the circle.
Q3. Show that the Hawaiian Earring is homeomorphic to the one-point compactification of the disjoint union of countably many copies of the open interval (0, 1).