Takehome Midterm
Math 351, Spring 2015
This midterm has two problems, each with four parts. Your solutions must be written up in
LATEX, and are due on Sunday, April 19 at midnight. Late solutions will not be accepted.
Here are the rules for the midterm:
• You may use the textbook as a reference, but you should not consult any other books
on topology.
• You should feel free to use the materials on the class web page, but you may not consult
any other internet sources.
• You may not discuss the problems or their solutions with anyone but me.
If you are having trouble, feel free to e-mail me a question, or come to my office hours
sometime this week.
1. Let X be a topological space. A subset A ⊂ X is said to be dense in X if the closure
of A is equal to X. (For example, the set Q of all rational numbers is dense in R.)
(a) If f : X → Y is a continuous surjection and A is dense in X, prove that f (A) is
dense in Y .
(b) Prove that the intersection of two dense open subsets of X is dense in X.
(c) Let X be a metric space, and suppose that X has a countable dense subset. Prove
that X has a countable basis for its topology.
(d) Let X = Rω under the product topology. Prove that X has a countable dense
subset.
2. Let X be a topological space. A subset A ⊂ X is said to be discrete in X if A has
no limit points in X.
(a) Prove that A is discrete in X if and only if A is closed in X and the topology
that A inherits as a subspace of X is the discrete topology.
(b) Give an example of a topological space X and a subset A ⊂ X such that the
subspace topology on A is the discrete topology but A is not discrete in X.
Justify your answer.
(c) Let X be a metric space and let A ⊂ X. Suppose there exists an > 0 so that
d(a1 , a2 ) ≥ for every pair of distinct points a1 , a2 ∈ A. Prove that A is discrete
in X.
(d) Let X = Rω under the uniform topology. Prove that X has an uncountable
discrete subset.