MATH 645, HOMEWORK #2
DUE MONDAY, MAY 29 AT 9AM.
Complete the exercise 0. and 5 of the remaining 6 exercises for full credit.
0. (Required) Let X be a topological space, and let A ⊂ X.
(a.) Prove that the closure of A in X, A, is the smallest closed subset of X that contains A.
I.e., prove that (i) A is closed, (ii) A ⊂ A, and (iii) if C ⊂ X is any closed subset having
A ⊂ C, then A ⊂ C.
(b.) Prove that the interior of A in X, IntA, is the largest open subset of X that is contained
within A. I.e., prove that (i) IntA is open, (ii) IntA ⊂ A, and (iii) if O ⊂ X is any open
subset having O ⊂ A, then O ⊂ IntA.
1. Prove that the order topology on R×R with respect to the dictionary order agrees with the
product topology Rd ×Rstd where Rd denotes R with the discrete topology and Rstd denotes R
with its standard topology.
2. Let L be a straight line in the Euclidean plane.
(a.) Describe the subspace topology on L when viewed as a subset of R`×Rstd where R` and
Rstd are as in problem 1.
(b.) Describe the subspace topology on L when viewed as a subset of R` × R`.
Hint: Consider cases: (i) L is horizontal; (ii) L is vertical; (iii) L has positive slope; and (iv)
L has negative slope.
3. Prove that a topological space X is Hausdorff if and only if ∆ = {(x, x) |x ∈ X} is a closed
subset of X ×X in the product topology.
4. Let A be a subset of a topological space X. Define the boundary of A in X to be
BdA = A ∩ (X \ A).
a. Prove that x ∈ BdA if and only if every neighborhood of x intersects both A and X \A.
b. Prove that A is the disjoint union of IntA and BdA. That is, show that A = IntA∪BdA
and IntA ∩ BdA = ∅.
c. Determine the boundary in R (with the standard topology) of the following subsets:
A1 = (0, 1], A2 = {1/n |n ∈ Z+}, and A3 = Q.
5. Determine the closure of the set K = {1/n |n ∈ Z+} in R with respect to each of the
following topologies:
a. the standard topology
b. the lower limit topology
c. the K-topology
d. the finite complement topology
e. the discrete topology
6. Show that if A is closed in X and B is closed in Y , then A×B is closed in X × Y with the
product topology.
1
2 DUE MONDAY, MAY 29 AT 9AM.
Suggested Additional Problems (do not turn in): Section 16: #3, 4, 6; Section 17: #6, 11, 20