MathDB

A subset

U \subset R

is open if for any

x \in U

, there exist real numbers

a, b

such that

x \in (a, b) \subset U

. Suppose

S \subset R

has the property that any open set intersecting

(0, 1)

also intersects

S

. Let

T

be a countable collection of open sets containing

S

. Prove that the intersection of all of the sets of

T

is not a countable subset of

R

. (A set

\Gamma

is countable if there exists a bijective function

f : \Gamma \to Z

Problem Statement