Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2015
Math 636 — Problem Set 4
Issued: 09.25
Due: 10.02
4.1. Show that the projection PY : X × Y −→ Y : (x, y) 7→ y is an open
map.
4.2. a) Let f : X −→ Y be a continuous map, and let Y be Hausdorff.
Show that the graph {(x, f (x)) : x ∈ X} is a closed subset of X × Y .
b) Show that a space X is Hausdorff if and only if the diagonal {(x, x) :
x ∈ X} is a closed subset of X × X.
4.3. Helly space Consider the space [0, 1][0,1] seen as the space of all functions f : [0, 1] −→ [0, 1] with the product topology. Show that the
subspace of all non-decreasing functions is compact.
4.4. Prove that product of connected spaces is connected.
4.5. Prove that for any finite discrete spaces A, B such that |A|, |B| ≥ 2
the spaces AN and B N are homeomorphic.
4.6. Proof of the Kuratowski Theorem Let X be a non-compact space,
and let {Fi }i∈I be a collection of closed subsets of X with finite intersection property
(intersection of any finite sub-collection is non-empty)
T
such that i∈I Fi = ∅. Consider the set Y = X ∪ {y0 }, where y0 ∈
/X
with the topology consisting of all subsets of X and sets of the form
{y0 } ∪ (Fi1 ∩ Fi2 ∩ · · · Fin ) ∪ K, where ik ∈ I and K ⊂ X.
Show that the projection map PY : X × Y −→ Y is not closed by
showing that PY (D) is not closed, where D is the closure in X × Y of
the set {(x, x) : x ∈ X}.