## Transcribed Text

TOPOLOGY
Question I.
Show that
defines a metric on IR2
•
Question 2.
Let (X, g) be a metric space.
Show that for all x, y, z EX
le(x, z) - g(y, z) I :S e(x, y).
Question 3.
Let A be a non-empty set of real numbers with infimum (greatest lower bound) b.
Prove that there is a sequence, (xn LeN> in A whose limit is b.
Question 4.
Let p be a prime number.
For each n E N* := N \ { 0}, let Vp ( n) be the exponent of p in the prime factorisation of n. This is
the highest power of P which divides n. In other words, vp(n) = j if and only if
(i) pi divides n, but
(ii) pi+l docs not divide n.
For each x E Q* := Q \ {O}, there are m, n EN* with Ix! := m. Define n
with !xi := as above.
Finally, define
A : ,n, X ,n, -► ]Ro+ , (x,y) f---7 {Q ( ) "'7' ",! ",!
p-Vp X-y
Show that (Q, dp) is a metric space.
if X = y
if X # y.
This is the p-adic metric on (Ql, which is important in number theory.
TOPOLOGY
Question 5
Let (X, e) be a metric space.
Prove that
g: Xx X -4 Rci, ( ) e(x,y) x,y -4 ( ) . 1 + g x ,y
is also a metric on X
Let (Y, a) be any metric space. Take functions f: X -4 Y and g: Y -4 X.
Prove that
(a) f is continuous with respect tog if and only if it is continuous with respect to (!;
(b) g is continuous with respect to l! if and only if it is continuous with respect to (l.
Question .J>
Let (X, e) be a metric space. Taking IR with its Euclidean metric, <'-, and X x X with one of the
canonical metrics on the Cartesian product.
Prove that
e: X x X -4 IRt, (x, y) i---+ e(x, y)
is continuous.
Given metric spaces, (X, e) and (Y, a), the function f: X -4 Y is uniformly continuous if and
only if given any c > 0 there is a J > 0 with a(i(u), f (v)) < c whenever e(u, v) < J.
Is e: X x X -> Rt uniformly continuous with respect to the Euclidean metric on JR and the
product metric on X x X?
Question}
Let E be the Euclidean metric on Rn and let A = [aij] nxn be a positive definite symmetric real
matrix. Define
(x,y) t---+ L %(Xi -yi)(xj - Yj)
i,j=l
where x =(xi, ... ,x.,) and y =(Yi, ... ,Yn)-
(a) Show that (Rn ,g) and (Rn ,<'-) arc isometric metric spaces.
(b) Show that the function
F: (Rn,€) -4 (Rn , g), X t---+ X
is continuous.
Question.
(i) Show that the complement of any finite subset of a metric space is open.
(ii) Show that every subset of a metric space is open if and only if each singleton subset is open.
TOPOLOGY
Question q
Prove that the function
max: JR x JR--+ JR, (x,y) -> max{x,y}
is continuous, where JR x JR and JR have their Euclidean metrics.
Question t. 0
Find a continuous bijection between topological spaces whose inverse is not continuous.
Question i. I
Let (X, T) be a topological space and take As::; X.
Prove that there is a maximal open subset of X disjoint from A.
This is the exterior of A, denoted cxt(A).
Question f. 2.
Let JR[t] denote the set of all real polynomials in the indeterminate t.
Let S be a subset of JR[t].
Define
V(S) := {x E JR I f(x) = 0 for every f ES}.
Define F := {V(S) I S JR[t]}.
Prove that there is a topology on JR, whose closed sets are precisely the elements of F and that
this topology is not metrisable.
This is the Zariski topology on JR and arises in algebraic geometry.
Question\ 1
Let X be a set.
Show that
TOPOLOGY
T := { A X I A = 0 or X \ A is finite } .
is a topology on X.
This is the finite complement topology on X. When is it metrisable?
Question I- 'j-
Let (X, 7), (Y,U) be topological spaces.
Take X x Y with its product topology.
Given b E Y, define
Endow im(ib ), the image of ib, with the topology induced by the product topology on X x Y.
Prove that
is a homeomorphism.
Question t· S
Let JR and JR2 have their respective Euclidean topologies. Take
s1 := { (x, y) E ]R2 I x2
+ y2 =- 1}
with the subspace topology induced from JR2
•
(i) Define the relation~ on JR by
a ~ b if and only if a - b E Z.
Prove that ~ is an equivalence relation on R
(ii) Let [a) be the ~-equivalence class containing a.
Put lR/z := { [a] I a E JR.}.
Endow IR/z with the quotient topology, that is, the topology induced by the natural projection
r,: IR R/z, a>--+ [a].
Prove that lR/z is homoeomorphic with S1
.
Question l'-
Take IR2 with its Euclidean topology.
TOPOLOGY
Prove that S1
= {(x, y) I x2 + y2
= l} c R2 is compact with respect to its subspace topology.
Question 1· 1
Let JI be the closed unit interval, [O, l] := {t E JR IO$ t $ l}, endowed with its Euclidean topology.
A path in the topological space (X, TI, from a EX to b EX is a continuous function
1:ll-+X
with 1(0) = a and ,(1) - b.
The subset A of the topological space (X, T) is path-connected if and only if for all a, b EA, there
is a path joining a to b in A.
(i) Prove that every path-connected set is connected.
(ii) Determine whether the subset
{(x,sin()) IO< x $ l}U {(0,y) 1-1 $ x $ I} X
of !R2 is
(a) connected,
(b) path-connected.
Question f. g
Let {(X).,U).) I>. EA } be a family of topological spaces.
Let TIX). be the (Cartesian) product of the sets X). (>.EA).
For each μ E A, let pr,., be the natural projection onto the μth factor
pr,.,. : II X). -+ Xμ , (xAhEA >---+ x,,,. AEA
The product topology on TIX). is the topology induced by {prμ Iμ EA}.
(i) Prove that G TIX). is open in the product topology if and only if
G= LJ Ga aEA
for some indexing set A, with each Ga of the form
Goe = II Ga). ACA
where Goe). an open subset of the X). and Goe). = X). for all but finitely many >.s.
(ii) Prove that for eachμ EA, the canonical projection,
prμ : Il X). ➔ Xμ
is an open mapping, that is, if G is an open subset of TIX). , then pr,.,.(G) is an open subset of X,,,.
TOPOLOGY
Question l- 'f
Recall that (X, T) is a T4 topological space if and only if given disjoint closed subsets, K and L
of X, there are disjoint open subsets, U and V, of X with
K <; U and L <; V
Prove that (X, T) is a T4 topological space if and only if given any closed subset F of X and any
open subset, Hof X with F <; H, there is an open subset, G, of X such that
FC GCGC H
Question l!O
Prove that every metric space is normal.
Question ..2, f
Let (X, e) and (Y, o-) be metric spaces.
Prove that if f: X --t Y is uniformly continuous, then it maps Cauchy sequences to Cauchy
sequences.
Question I!. '2..
Let (X, e) and (Y, o-) be metric spaces.
Prove that if X is compact, every continuous function f: X --t Y is uniformly continuous.

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