The problems are due on November 4.
Problem 1.
Given a set 
, we say that x is an interior point of A,
if A is a neighborhood for x. We let int(A), the interior of
A, be the set of all interior points of A.
Show that the interior of a set
is the union of all open sets contained in the set:
Problem 2. p. 109, Problem 39.
Problem 3.
Let 
be a decreasing sequence of non-empty
compact sets, i.e., 
for all 
.
Show that
Problem 4.
Find a decreasing sequence of non-empty
closed sets, so that
Problem 5.
Show that the function 
is continuous on 
, if and only if the pre-image 
is open for all open sets 
.