UNIVERSITY OF WAIKATO
Department of Mathematics
MATH501-08B – Metric Spaces
TEST
Friday 17 October 2008 - (50 mins) – Attempt 6 questions
1. Prove that if f :
[ ]
a,b !R is Riemann integrable, then for every ! >0 there exists a partition P! of[ ]
a,b such that U(f,P!)"L(f,P!)<!.2. Prove that if f :
[ ]
a,b !R is Riemann integrable and c>0 then c.f is Riemann integrable and c.fa
!
b =c a f!
b .3. Prove that if f :
[ ]
a,b !R is continuous then it is uniformly continuous.4. Let
(
fn:n!N)
be a sequence of continuous functions, and suppose that fn! f uniformly on a,b( )
. Show that f is continuous on( )
a,b .5. Let (X,!) be a compact topological space and let A! X be closed. Show that A is a compact subset.
6. Let (X,!) be a compact topological space and let f :X!Y be continuous, where
(
Y,!)
is atopological space. Prove that f (X) is a compact subset of Y.
7. Let (X,!) be a topological space and A! X a connected subset. Prove that the closure A is also connected.
- 2 -
8. Let (X,!) be a topological space and A! X a non-empty subset. Let x!X be such that ! open P with x!P,P"A#$. Show that x!A.
9. Let (X,!) be a metrizable topological space. Prove that the topology is Hansdorff, then that it is normal.
10. Let R have its usual topology. Show that the interval (a,b)!R, with a<b, is connected.
11. Let f :Rn!Rm be differentiable at a!Rn. Prove that f is continuous at a.
12. Define equivalent norms on a vector space V over R. Show that if V =Rn, all norms are equivalent to
x 0 = max
1!i!n
