Igor Zelenko, Fall 2009 1
due to Sept 14, 2009
1. Let X be a topological space. Recall that a set A ⊂ X is said to be nowhere dense in X if int( ¯ ) = ∅ .
a. Let U ⊂ X be open. Show that the boundary ∂U of U is closed and nowhere dense in X .
b. Conversely, show that every closed, nowhere dense set is the boundary of an open set.
c. Is ∂U necessarily nowhere dense, if we drop the assumption that U is open?
2. An open subset U in a topological space is said to be regularly open if U is the interior of its closure.
A closed set is regularly closed if it is the closure of its interior. Show: a. The complement of a regularly open set is regularly closed, and vice versa; b. There are open sets in
R
(with the usual Euclidean topology) which are not regularly open; c. If A is any subset of a topological space, then int( ¯ ) (the interior of the closure of A ) is regularly open.
3. Let B be the collection of all sets U
ε,L
1
,...,Lr ( x )
( x ) ⊂
R
2 corresponding to all x ∈
R
2 , ε > 0, and finite collections of lines L
1
, . . . , L r passing through x , where U
ε,L
1
,...,Lr ( x )
( x ) = B
ε
( x ) \{ L
1
, . . . , L r
} ∪{ x } .
a. Show that B forms a basis for a topology on
R
2 (Sliced Pie Space topology); b. Given any line L in
R
2 , describe the subspace topology that one gets on L from this new topology; c. Show that Sliced Pie topology is not first countable.
4. Let
R ` denote the real numbers under the lower limit topology ( i.e. the topology with the basis consisting of all left-closed, right-open intervals [ a, b )). Describe the closure in
R ` of the following sets: a.
X = { 1 n
: n ∈
N
} ; b.
X = {− 1 n
: n ∈
N
} .
5. Let
R
2
Z denote the plane with the Zariski topology . A basis B for
R
2
Z consists of all subsets of
R
2 whose complements are zero sets of polynomials with real coefficients. That is, U is a basic open set for
R
2
Z
, if there is a polynomial p :
R
2 →
R with real coefficients such that U =
R
2 \{ ( x, y ) : p ( x, y ) = 0 } .
a. Show that B is a basis for a topology; b. Let f ( x, y ) and g ( x, y ) be two polynomials with real coefficients, and define a function F :
R
2
Z
→
R
2
Z by F ( x, y ) = f ( x, y ) , g ( x, y ) . Show that F is continuous in the Zariski topology.