Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2015
Math 636 — Problem Set 1
Issued: 09.04
Due: 09.11
1.1. Prove that an open set U has a non-empty intersection with a set A
if and only if it has a non-empty intersection with its closure A.
1.2. Show that intersection of two open dense sets is dense.
1.3. Open domains. A subset A of a topological space is called an open
domain if A = Int(A).
a) Give an example of an open subset of R that is not an open domain.
b) Prove that interior of a closed set is an open domain.
c) Prove that intersection of two open domains is an open domain.
Is it true that union of two open domains is an open domain?
1.4. Prove that a metric space is second countable if and only if it is separable. (A topological space is called separable if it has a countable
dense subset.)
1.5. Sorgenfrey line. Consider the set of half-open intervals {[a, b) : a <
b} of the real line, where [a, b) = {x ∈ R : a ≤ x < b}.
a) Show that it is a basis of a topology on R. We call it Sorgenfrey
line.
b) Prove that this topology is first countable, but not second countable.
c) Show that Sorgenfrey line is separable, hence it is not homeomorphic to any metric space.
1.6. Kuratowski’s problem. Prove that by applying alternatively closure
and complement operators to a subset A of a topological space X one
obtains at most 14 different sets. Find a subset of the real line from
which one obtains in this way exactly 14 different sets. (Hint: show
0 0 0
0
that A− − − − = A− − , where A− is closure and A0 is complement.)