Download Topology Masters Comp

Topology Masters Comp
March 23, 2005
1. Let X be a topological space and let A1 , A2 , · · · be subsets of X. Prove
or give a counter example (A = closure of A):
(a) A1 ∪ A2 = A1 ∪ A2
(b) A1 ∩ A2 = A1 ∩ A2
(c)
∞
S
n=1
An =
∞
S
An
n=1
2. Let R
l be the real numbers with the usual topology. Prove or give a
counterexample:
(a) If A and B are compact subsets of R
l then
A + B ≡ {a + b : a ∈ A, b ∈ B} is a compact subset of R
l
(b) If A and B are connected subsets of R
l then A + B is a connected
subset of R
l.
3. Let X be a Hausdorff space and suppose that h : X → X is a homeomorphism and g : X → X is continuous. Prove that
{(h(x), g(x)) : x ∈ X} is a closed subset of X × X (with the product
topology).
1