Download Topology III Exercise set 6 1. Show that the following are equivalent

Topology III
Exercise set 6
1. Show that the following are equivalent for a topological space X:
A. X is normal.
B. Every finite open cover of X has a finite open point-star refinement.
C. Every locally finite open cover of X has a locally finite open point-star refinement.
2. (a) Show that, for every open cover U of the ordinal space [0, ω1 ), there exists α < ω1
such that [α, ω1 ) ⊂ St(α, U ).
(b) Show that [0, ω1 ) is not fully normal.
3. (a) Let X be a compact Hausdorff space and let F be a closed Gδ -subset of K. Show
that F has a countable nbhd base in X.
(b) Show that a compact Hausdorff space X is metrizable iff the diagonal ∆ = {(x, x) :
x ∈ X} is a Gδ -set in the space X 2 .
4. For every A ⊂ I, let Ai = A × {i} for i = 0, 1. The two arrows space K is the
set [0, 1)0 ∪ (0, 1]1 equipped with the topology which has a base formed by all sets
[a, b)0 ∪ (a, b)1 and (a, b)0 ∪ (a, b]1 , where 0 ≤ a < b ≤ 1.
Show that p : (x, i) 7→ x is a perfect map K → I and deduce that K is compact.
5. Let K and p be as in Problem 4.
(a) Show that for every F ⊂c K, the set p−1 (p(F )) r F is countable.
(b) Show that K is perfectly normal.
(c) Show that K 2 is not perfectly normal.
6. A pseudocompact space is a Tihonov space on which every continuous real-valued
function is bounded. Show that the following are equivalent for a Tihonov space X:
A. X is pseudocompact.
B. Every continuous pseudometric of X is bounded.
C. Every continuous pseudometric of X is totally bounded.