Topics in Advanced Mathematics: Topology
Homework 7b
Wojciech Komornicki
3 April 2014
3.4 1 In the following, X is a subspace of R2 and x0 , x1 are points of X. Write down, if possible, explicit paths in
X joining x0 to x1 .
(i) X = (x, y) R2 | |x| > 1 or
Topics in Advanced Mathematics: Topology
Homework 5a
Solutions
4 March 2014
2.6 4 Prove that subspaces and (nite) products of spaces satisfying te rst axiom of countabillity also satisfy the
rst axiom of countability.
Proof: Let X satisfy the rst axiom of
Topics in Advanced Mathematics: Topology
Homework 8
Soluions
10 April 2014
3.4 8 Let X1 , X2 be subsets of X such that X = IntX1
component of X1 meets X2 .
IntX2 . Prove that, if X is path-connected, then each path
Proof: Since X is path-connected, it is
Topics in Advanced Mathematics: Topology
Homework 4b
Wojciech Komornicki
25 Febrauary 2014
2.5 1 Prove the continuity of the following functions : R3 R.
(i) (x, y, z) P (x, y, z) where P is a polynomial.
Proof:
We rst show that x P (x) where P is a polyno
Topics in Advanced Mathematics: Topology
Homework 10
Wojciech Komornicki
01 May 2014
4.4 1 Let x0 Sn and let be the hyperplane in Rn+1 which is perpendicular to the line determined by the origin
and x0 and which passes through the origin. Let s : Sn \ cfw
Topics in Advanced Mathematics: Topology
Homework 4a
Wojciech Komornicki
25 Febrauary 2014
2.4 1 Prove that the relation X is a subspace of Y is a partial order relation of topological spaces.
Proof:
Reexive Let X be a topological space. If A X is open th
Topics in Advanced Mathematics: Topology
Examination I
Solutions
11 March 2014
I. (36 points) Denitions Complete the following denitions:
a) Let X be a set and N X
a)
b)
c)
d)
(X). N is called a neighbourhood topology on X if
for every x X and N N (x), x
Topics in Advanced Mathematics: Topology
Homework 6a
Wojciech Komornicki
20 March 2014
2.8 2 Let f : R0 R0 be a continuous function such that
(i) f (x) = 0 x = 0,
(ii) x x f (x) f (x ),
(iii) f (x + x ) f (x) + f (x )
Let d be a metric on X. Show that the
Topics in Advanced Mathematics: Topology
Examination II
Solutions
17 April 2014
I. (24 points) Denitions Complete the following denitions:
a) Let X be a set. A metric d on X is a function d X X
R such that
i. d(x, y) 0 for all x, y X; d(x, y) = 0 if and
Topics in Advanced Mathematics: Topology
Homework 3b
Wojciech Komornicki
20 Febrauary 2014
2.3 1 State which of the following sets of points (x, y) of R2 are (a) open, (b) closed, (c) neither open nor closed.
(i)
(x, y) R2 : |x| < 1 and |y| < 1
Solution
T
Topics in Advanced Mathematics: Topology
Examination III
Solutions
13 May 2014
I. (24 points) Denitions Complete the following denitions:
a) Let X be a topological space. X is said to be compact if every open cover of X has a nite
subcover.
b) Let f X Y b
Topics in Advanced Mathematics: Topology
Homework 1 Solutions
Wojciech Komornicki
6 February 2014
1.1 1 Let a R and N be a subset of R. Prove that the following conditions are equivalent:
(a) N is a neighbourhood of a
(b) There is a > 0 such that [a , a +
Topics in Advanced Mathematics: Topology
Homework 3a Solutions
Wojciech Komornicki
18 Febrauary 2014
2.2 3 Let X be a topological space, and let A X. A point x in X is called limit point of A if each neighbourhood
of x contains points of A other than x. T
Topics in Advanced Mathematics: Topology
Homework 2b Solutions
Wojciech Komornicki
13 February 2014
2.1 1 Let X be a set with a neighbourhood topology. Prove that if A and B are subsets of X then,
IntA IntB = Int(A B)
IntA IntB Int(A B)
Proof: First, we s
Topics in Advanced Mathematics: Topology
Homework 2a Solutions
Wojciech Komornicki
11 February 2014
1.3 1 Prove that the Cantor set is uncountable.
Proof: The function dened in problem 1.3 2 is surjective. Since I is uncountable, K is uncountable.
1.3 2 P