(1) Let X be a topological space; let A be a subset of X. Suppose that for each x ∈ A there is an 10%
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) Let S1=©
(x, y) ∈ R2| x2+ y2= 1ª
, and let f : S1→ R be a continuous function. Prove that 10%
there exists a point z of S1 such that f (z) = f (−z).
(4) Show that if d is a metric for X, then 10%
d1(x, y) = d(x, y) 1 + d(x, y) is a bounded metric for X.
(5) Show that a subspace of a closed normal space is normal.
10%
(6) Let f, g : X → Y be continuous; assume that Y is Hausdorff. Show that {x|f (x) = g(x)} is 10%
closed in X.
(7) Let f : X → Y be continuous; let X be compact. Prove that the set Z = f (X) is compact in 10%
Y .
(8) Consider the product space X × Y , where Y is compact. Suppose that N is an open set of 10%
X × Y containing the slice x0× Y of X × Y , prove that N contains some tube W × Y about x0× Y , where W is a neighborhood of x0 in X.
(9) Show that if X is regular, every pair of points of X have neighborhoods whose closures are 10%
disjoint.
(10) Show that if U is an open connected subspace of R2, then U is path connected. [Hint: Show 10%
that given x0∈ U , the set of points that can be joined to x0by a path in U is both open and closed in U .]
