(4 marks) Let (X, d) be a metric space and A ⫃ X a nonempty subset of X. Prove that a point is an interior point of A, x € Aº if and only if
dist (r, Ac) > 0,
where
dist (x, Ac) = inf{d(x,y) | ye Ac}.
(4 marks) Let X = {a,b,c} be a three-point space equipped with the topology Tx = {Ø, {a}, {b}, {a,b}, {a,b,c}}, and let Y = {0, 1} be a two-point space with the topology Ty = {Ø, {1}, {0, 1}}. Define a map f: (X, Tx) (Y, Ty) by
f(a)=1, f(b) = 0, f(c) = 0.
Let g: (Y, Ty) → (X, Tx) be defined by
g(0) = a; g(1) = b.
Are the maps f and g continuous with respect to the stated topologies? Provide reasons for your answer.
(4 marks) Let the set X = [0, 2] u {3} be equipped with the subspace topology inherited from the standard metric topology on the reals R.
(i) Is the set [0, 1) open in the subspace topology on X?
(ii) Is the set {3} open in the subspace topology on X?
Provide reasons for your answers.
(4 marks) All spaces below are equipped with the standard metric topology inherited from R.
(i) Is the space X = (-3, 2) homeomorphic to Y = [0, 1]?
(ii) Is the space X = [0, 2]u[4, 5] homeomorphic to Y = [10, 15]?
Provide reasons for your answers.