Download Math 636 — Problem Set 6 Issued: 10.16 Due: 10.23

Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2015
Math 636 — Problem Set 6
Issued: 10.16
Due: 10.23
6.1. Let X be a normal space. Show that a closed subset A ⊂ X is
intersection of a countable set of open sets (i.e., is a Gδ -set) if and
only if there exists a continuous function f : X −→ [0, 1] such that
f −1 (0) = A.
6.2. A topological space X is perfectly normal if every closed set F ⊂ X is
equal to intersection of a countable set of open sets.
(a) Prove that every metric space is perfectly normal.
(b) Prove that the direct product [0, 1]S , where S is uncountable, is not
perfectly normal.
6.3. Let X be a completely regular space. Let A ⊂ X be compact, and
let B ⊂ X be closed. Show that there exists a continuous function
f : X −→ [0, 1] such that f |A = 0 and f |B = 1.
6.4. Let X be a Hausdorff space, and suppose that Y ⊂ X is dense and
locally compact. Show that Y is open.
6.5. Partition of unity Let X be a compact Hausdorff space, and let
{Ui }ni=1 be a finite open cover of X. Show that there exist continuous functions φi P
: X −→ [0, 1] such that φi |X\Ui = 0 for every
i = 1, 2, . . . , n, and ni=1 φi = 1.