(1) Let X be a topological space; let A be a subset of X. Suppose that for each x ∈ A there is an 8%
open set U containing x such that U ⊂ A. Show that A is open.
(2) Show that if A is a basis for a topology on X, then the topology generated by A equals the 10%
intersection of all topologies on X that contain A.
(3) A map f : X → Y is said to be an open map if for every open set U of X, the set f (U ) is open 8%
in Y . Show that π1: X × Y → X is an open map.
(4) Let X be an ordered set in the order topology. Show that (a, b) ⊂ [a, b].
8%
(5) In the finite complement topology Tf = { U | R \ U is either finite or is all of R } on R, to what 8%
point or points does the sequence xn= 1/n converge?
(6) Show that X is Hausdorff if and only the diagonal ∆ = { x × x | x ∈ X } is closed in X × X.
10%
(7) If A ⊂ X, we define the boundary of A by the equation 16%
BdA = ¯A ∩ (X \ A).
(a) Show that IntA and BdA are disjoint, and ¯A = IntA ∪ BdA.
(b) If U is open, is it true that U = Int( ¯U )? Justify your answer. [Hint: You may find the following fact useful. U is open if and only if BdU = ¯U \ U .]
(8) Suppose that f : X → Y is continuous. If x is a limit point of the subset A of X, is it necessarily 8%
true that f (x) is a limit point of f (A)?
(9) Show that the subspace (a, b) of R is homeomorphic with (0, 1).
8%
(10) Let X be a metric space with metric d. Show that d : X × X → R is continuous.
8%
(11) Let A ⊂ X. If d is a metric for the topology of X, show that d|A × A is a metric for the 8%
subspace topology on A.
