221: Analysis
1
Review of real analysis
2
Metric spaces
Definition A metric space is a pair (X, d) consisting of a non-empty set X
and a map d : X × X → R such that for all x, y, z ∈ X,
(i) d(x, y) ≥ 0
(ii) d(x, y) = 0 if and only if x = y
(iii) d(x, y) = d(y, x)
(iv) d(x, z) ≤ d(x, y) + d(y, z) (the Triangle Inequality).
d is called a metric or distance function.
Example
(i) The discrete metric: X a non-empty set, d(x, y) =
0
1
(ii) Euclidean spaces: Rn with the Euclidean metric
p
d((x1 , . . . , xn ), (y1 , . . . , yn )) = (x1 − y1 )2 + · · · + (xn − yn )2 .
(iii) Other metrics on R2 :
(a) d1 ((x1 , x2 ), (y1 , y2 )) = |x1 − y1 | + |x2 − y2 |
(b) d∞ ((x1 , x2 ), (y1 , y2 )) = maxi=1,2 |xi − yi |
(iv) The function space C([a, b]) with
(a) d(f, g) = supa≤x≤b |f (x) − g(x)|
Rb
(b) d(f, g) = a |f (x) − g(x)|dx
(v) The sequence space c0 with d((xn ), (yn )) = supn |xn − yn |.
1
if x = y
.
if x =
6 y
(vi) Subspaces: If (X, d) is a metric space and A is a subset of X then (A, dA )
is a metric space where dA (x, y) = d(x, y) for all x, y ∈ A. (A, dA ) is
called a subspace of (X, d) and dA is called the induced metric.
(vii) Product spaces: If (X1 , d1 ) and (X2 , d2 ) are metric spaces then each of
the following is a metric on X1 × X2 ,
(a) d((x1 , x2 ), (y1 , y2 )) = d1 (x1 , y1 ) + d2 (x2 , y2 )
p
(b) d((x1 , x2 ), (y1 , y2 )) = d1 (x1 , y1 )2 + d2 (x2 , y2 )2
(c) d((x1 , x2 ), (y1 , y2 )) = max{d1 (x1 , y1 ), d2 (x2 , y2 )}
Lemma 2.1. (Cauchy-Schwarz Inequality) Let x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) ∈
Rn . Then
v
v
u n
u n
n
X
uX uX
2t
t
|xi ||yi | ≤
xi
yi2 .
i=1
i=1
i=1
Definition Let (X, d) be a metric space. For each x ∈ X and each r > 0
define
(i) the sphere with centre x and radius r, S(x, r) = {y ∈ X : d(x, y) = r},
(ii) the open ball with centre x and radius r, B(x, r) = {y ∈ X : d(x, y) < r},
(iii) the closed ball with centre x and radius r, B[x, r] = {y ∈ X : d(x, y) ≤ r}.
Lemma 2.2. Let B(a, r) be an open ball in a metric space (X, d). If x ∈
B(a, r) then B(x, r0 ) ⊆ B(a, r) for all 0 < r0 ≤ r − d(x, a)
Definition Let (X, d) be a metric space and let A be a subset of X. The
distance from a point x ∈ X to A is defined as
d(x, A) = inf d(x, y).
y∈A
Lemma 2.3. Let B(a, r) be an open ball in a metric space (X, d). If x ∈ X
and x ∈
/ B(a, r) then d(x, B(a, r)) ≥ d(a, x) − r.
Definition Let (X, d) be a metric space and let A be a subset of X. Define
the diameter of A to be
δ(A) = sup d(x, y)
x,y∈A
or if this supremum does not exist then set δ(A) = ∞.
Lemma 2.4. Let B[a, r] be a closed ball in a metric space (X, d). Then the
diameter of B[a, r] is ≤ 2r.
2
Definition Let (X, d) be a metric space and let A be a subset of X. Then A
is called a bounded set if there exists a closed ball B[x, r] with A ⊆ B[x, r].
Proposition 2.5. Let (X, d) be a metric space and let A be a subset of X.
Then A is bounded iff A has finite diameter.
Proposition 2.6. Let (X, d) be a metric space. The union of a pair of bounded
sets is bounded.
Definition Let (X, d) be a metric space and let A be a subset of X. Then A
is called an open set if for each x ∈ A there exists an open ball B(x, r) with
B(x, r) ⊆ A.
Theorem 2.7. Let (X, d) be a metric space. Every open ball in (X, d) is an
open set.
Theorem 2.8. Let (X, d) be a metric space and let A be a subset of X. Then
A is an open set iff A is a union of open balls.
Theorem 2.9. Let (X, d) be a metric space. Then
(i) ∅ and X are open sets,
(ii) the union of any collection of open sets is an open set,
(iii) the intersection of finitely many open sets is an open set.
Definition Let (X, d) be a metric space and let A be a subset of X. A point
x ∈ A is called an interior point of A if there exists an open ball B(x, r) with
B(x, r) ⊆ A. The interior of A, denoted int(A), is the set of all interior points
of A.
Theorem 2.10. Let (X, d) be a metric space and let A be a subset of X. Then
the interior of A is an open set.
Example In R, every open interval (a, b) is an open set with respect to the
Euclidean metric. Intervals of the form (a, b], [a, b), [a, b] are not open sets. A
set {x} consisting of a single point is not an open set. The interior of [a, b] is
(a, b).
Definition Let (X, d) be a metric space. A sequence in X is a mapping
s : N → X and is usually written as (xn ) where xn = s(n) for each n ∈ N.
A sequence (xn ) is said to converge to a point x ∈ X if given any > 0
there exists N ∈ N such that d(xn , x) ≤ for all n ≥ N . The point x is called
the limit of the sequence and we write limn→∞ xn = x.
A sequence (xn ) is said to be bounded if the set {xn : n ∈ N} is a bounded
set in (X, d).
3
Theorem 2.11. Let (X, d) be a metric space. A convergent sequence in (X, d)
is bounded and has a unique limit.
Definition Let (X, d) be a metric space and let A be a subset of X. A point
x ∈ X is called a limit point of A if there exists a sequence (xn ) in A\{x}
which converges to x.
Theorem 2.12. Let (X, d) be a metric space and let A be a subset of X. A
point x ∈ X is a limit point of A iff A\{x} ∩ B(x, ) is non-empty for each
open ball B(x, ).
Definition Let (X, d) be a metric space and let A be a subset of X. Then A
is called a closed set if it contains all of its limit points.
Theorem 2.13. Let (X, d) be a metric space. Then every closed ball in (X, d)
is a closed set in (X, d).
Theorem 2.14. Let (X, d) be a metric space and let A be a subset of X. Then
A is a closed set iff X\A is an open set.
Theorem 2.15. Let (X, d) be a metric space. Then
(i) ∅ and X are closed sets,
(ii) the intersection of any collection of closed sets is a closed set,
(iii) the union of finitely many closed sets is a closed set.
Example In R, every closed interval [a, b] is a closed set with respect to the
Euclidean metric. Intervals of the form (a, b], [a, b), (a, b) are not closed sets.
A set {x} consisting of a single point is a closed set.
Definition Let (X, d) be a metric space and let A be a subset of X. The
closure of A, denoted Ā, is the union of A and the set of limit points of A.
A is said to be dense in X if Ā = X. The boundary of A is ∂A = Ā ∩ X\A.
Theorem 2.16. Let (X, d) be a metric space and let A be a subset of X. Then
the closure of A is a closed set.
Example In R with the Euclidean metric, the closure of each of the intervals
(a, b], [a, b) and (a, b) is [a, b]. The boundary of each of the intervals (a, b], [a, b)
and (a, b) is {a, b}. The closure of R\{0} is R.
Definition Let (X, d) be a metric space. A sequence (xn ) in X is called a
Cauchy sequence if given any > 0 there exists N ∈ N such that
d(xm , xn ) < for all m, n ≥ N.
4
Theorem 2.17. Let (X, d) be a metric space.
(i) Every convergent sequence in (X, d) is a Cauchy sequence in (X, d).
(ii) Every Cauchy sequence in (X, d) is bounded.
Definition A metric space (X, d) is called complete if every Cauchy sequence
(xn ) in X converges to a point in X.
Example Consider (0, 1] with metric induced from the Euclidean metric on
R. The sequence ( n1 ) is a Cauchy sequence in (0, 1] but does not converge to a
point in (0, 1] so (0, 1] is not complete.
Proposition 2.18. A sequence in Rm converges with respect to the Euclidean
metric iff each coordinate sequence in R converges with respect to the Euclidean
metric.
Proposition 2.19. Rm with the Euclidean metric is complete.
Definition Let (X, d) be a metric space. A mapping T : X → X is called a
contraction mapping if there exists a real number α with 0 < α < 1 such that
d(T x, T y) ≤ αd(x, y) for all x, y ∈ X.
Theorem 2.20. (Banach’s Fixed Point Theorem) A contraction mapping T :
X → X on a complete metric space (X, d) has a unique fixed point. (i.e. there
exists exactly one element x ∈ X such that T (x) = x).
Example Let f : [a, b] → [a, b] be a differentiable function. If there exists
K < 1 such that |f 0 (x)| ≤ K for all x ∈ [a, b] then f is a contraction mapping
on [a, b] with respect to the Euclidean metric. The closed interval [a, b] is
complete with respect to the induced metric and so by Banach’s fixed point
theorem, f has a unique fixed point in [a, b].
Theorem 2.21. Let (X, d) be a metric space and let A be a subset of X.
(i) If (A, dA ) is complete then A is a closed set.
(ii) If (X, d) is complete and A is a closed set then (A, dA ) is complete.
Definition Let X be a non-empty set and let d, d0 be two metrics on X.
Then d, d0 are called topologically equivalent if for every sequence (xn ) in X,
(xn ) converges to a point x in (X, d) iff (xn ) converges to x in (X, d0 ).
Theorem 2.22. Let X be a non-empty set and let d, d0 be two metrics on X.
The following are equivalent:
(i) d, d0 are topologically equivalent metrics,
5
(ii) A set is closed in (X, d) iff it is closed in (X, d0 ),
(iii) A set is open in (X, d) iff it is open in (X, d0 ).
Definition Let X be a non-empty set and let d, d0 be two metrics on X. Then
d, d0 are called Lipschitz equivalent if there exist real numbers m1 , m2 > 0 such
that
m1 d(x, y) ≤ d0 (x.y) ≤ m2 d(x, y) for all x, y ∈ X.
Proposition 2.23. Let X be a non-empty set and let d, d0 be two metrics on X.
If d, d0 are Lipschitz equivalent metrics then they are topologically equivalent
metrics.
Proposition 2.24. Let X be a non-empty set and let d, d0 be two metrics on
X. If d, d0 are Lipschitz equivalent metrics then
(i) (xn ) is a Cauchy sequence in (X, d) iff (xn ) is a Cauchy sequence in
(X, d0 ),
(ii) A is a bounded set in (X, d) iff A is a bounded set in (X, d0 ),
(iii) (X, d) is complete iff (X, d0 ) is complete.
Example On R2 , the Euclidean metric d and the bounded metric d0 (x, y) =
d(x,y)
are topologically equivalent but not Lipschitz equivalent.
1+d(x,y)
Definition Let (X, d) and (Y, d0 ) be metric spaces. A mapping T : X → Y
is a called continuous at a point x0 ∈ X if given any > 0 there exists δ > 0
such that
d(x, x0 ) < δ =⇒ d0 (T x, T x0 ) < .
T is called continuous if it is continuous at every point of X.
Theorem 2.25. Let (X, d) and (Y, d0 ) be metric spaces. A mapping T : X →
Y is a continuous mapping iff for every sequence (xn ) converging to x in (X, d),
the sequence (T xn ) converges to T x in (Y, d0 ).
Example With respect to the Euclidean metric,
(i) (addition) p : R2 → R, p(x, y) = x + y is a continuous mapping,
(ii) (multiplication) q : R2 → R, q(x, y) = xy is a continuous mapping,
(iii) (coordinate projections) pj : Rm → R, pj (x1 , . . . , xm ) = xj is a continuous
mapping for each j = 1, . . . , m.
Theorem 2.26. The composition of continuous mappings is a continuous
mapping.
6
Theorem 2.27. Let X be a non-empty set and let d1 , d2 be two metrics on X.
Let (Y, d) be a metric space. Then the following are equivalent:
(i) d1 , d2 are topologically equivalent metrics,
(ii) a mapping T : Y → X is (d, d1 )-continuous iff it is (d, d2 )-continuous,
(iii) a mapping T : X → Y is (d1 , d)-continuous iff it is (d2 , d)-continuous,
Definition Let (X, d) and (Y, d0 ) be metric spaces. A mapping T : X → Y
is called an isometry if d0 (T x, T y) = d(x, y) for all x, y ∈ X. (i.e. T preserves
distances).
Two metric spaces (X, d) and (Y, d0 ) are called isometric if there exists an
isometry T : X → Y which is surjective.
Proposition 2.28. Let (X, d) and (Y, d0 ) be metric spaces and let T : X → Y
be an isometry. Then T is one-to-one, continuous and T −1 : T (X) → X is
continuous.
Theorem 2.29. Let (X, d) and (Y, d0 ) be metric spaces. Then a mapping
T : X → Y is continuous iff the preimage T −1 (U ) = {x ∈ X : T x ∈ U } is
open in (X, d) for each U open in (Y, d0 ).
Definition Let (X, d) and (Y, d0 ) be metric spaces. A mapping T : X → Y is
called uniformly continuous if given any > 0 there exists δ > 0 such that
d(x, y) < δ =⇒ d0 (T x, T y) < .
Theorem 2.30. Let (X, d) and (Y, d0 ) be metric spaces. If a mapping T :
X → Y is uniformly continuous then T sends Cauchy sequences in (X, d) to
Cauchy sequences in (Y, d0 ).
Example (i) f : R\{0} → R, f (x) =
continuous.
1
x
is continuous but not uniformly
(ii) f : R → R, f (x) = x2 is continuous but not uniformly continuous.
(iii) Contraction mappings are uniformly continuous.
(iv) Isometries are uniformly continuous.
Proposition 2.31. Let (X, d) be a metric space. A mapping f : X → Rm ,
f = (f1 , . . . , fm ) is continuous iff each component function fj : X → R is
continuous.
Proposition 2.32. Let (X, d) be a metric space. Let f : X → R and g : X →
R be continuous functions. Then f +g, f.g are continuous and fg is continuous
wherever it is defined.
7
Definition Let (X, d) and (Y, d0 ) be metric spaces. Let (fn ) be a sequence of
functions fn : X → Y . Then (fn ) is said to converge pointwise to a function
f : X → Y if for each x0 ∈ X, given any > 0 there exists N ∈ N such that
d0 (fn (x0 ), f (x0 )) < for all n ≥ N.
The sequence (fn ) is said to converge uniformly to f : X → Y if given any
> 0 there exists N ∈ N such that
d0 (fn (x), f (x)) < for all n ≥ N, and for all x ∈ X.
Theorem 2.33. (Uniform Limit Theorem) Let (X, d) and (Y, d0 ) be metric
spaces. Let (fn ) be a sequence of continuous functions fn : X → Y . If (fn )
converges uniformly to f : X → Y then f is continuous.
1
Example For each n ∈ N let fn : (0, ∞) → R, fn (x) = nx
. Then the sequence
(fn ) converges pointwise but not uniformly to the zero function f : (0, ∞) → R,
f (x) = 0.
Lemma 2.34. Let (X, d) be a metric space.
(i) |d(x, z) − d(y, u)| ≤ d(x, y) + d(z, u) for all x, y, z, u ∈ X.
(ii) |d(x, z) − d(y, z)| ≤ d(x, y) for all x, y, z ∈ X.
˜ is called a
Definition Let (X, d) be a metric space. A metric space (X̃, d)
completion for (X, d) if there exists an isometry i : X → X̃ such that
(i) i(X) is dense in X̃,
˜ is a complete metric space.
(ii) (X̃, d)
˜ .
Theorem 2.35. Every metric space (X, d) has a completion (X̃, d)
Example With the Euclidean metric, R is a completion of Q.
8
3
Topological Spaces
Definition A topological space is a pair (X, τ ) consisting of a set X and a
collection τ of subsets of X such that
(i) ∅ and X are in τ ,
(ii) if (Uα ) is a collection of sets in τ then
S
Uα is in τ ,
T
(iii) if U1 , . . . , Un is a finite collection of sets in τ then nj=1 Uj is in τ .
α
The collection τ is called a topology on X. The elements of τ are called open
sets.
Example (i) The trivial topology: Let X be a non-empty set and let τ =
{∅, X}.
(ii) The discrete topology: Let X be a non-empty set and let τ be the collection of all subsets of X.
(iii) The metric topology: Let (X, d) be a metric space and let τ be the collection of all open sets in (X, d).
(iv) The metric topology given by the Euclidean metric on Rn is called the
usual topology on Rn .
(v) The Sierpinski space: Let X = {a, b} and let τ = {∅, {a}, X}.
(vi) The Zariski topology: Let X be a non-empty set and let τ be the collection
of all subsets U such that X\U is a finite set of points together with the
empty set ∅.
Definition A topological space (X, τ ) is called metrizable if there exists a
metric d on X such that the metric topology on (X, d) is the same as the
topology τ .
Example (i) If X is a set containing at least two elements then the trivial
topology on X is not metrizable.
(ii) The Sierpinski space and the Zariski topology are not metrizable.
Definition A subset A of a topological space (X, τ ) is called a closed set if
its complement X\A is an open set.
Theorem 3.1. Let (X, τ ) be a topological space. Then
(i) ∅ and X are closed sets,
(ii) if (Uα ) is a collection of closed sets then
9
T
α
Uα is a closed set,
(iii) if U1 , . . . , Un is a finite collection of closed sets then
set.
Sn
j=1
Uj is a closed
Definition Let τ and τ 0 be two topologies on a non-empty set X. If τ ⊆ τ 0
then τ 0 is said to be finer than τ . We also say that τ is coarser than τ 0 .
Example Let X = {a, b} and let τ1 = {∅, X}, τ2 = {∅, {a}, X} and τ3 =
{∅, {a}, {b}, X}. Then τ3 is finer than τ2 and τ2 is finer than τ1 .
Definition Let X be a non-empty set. A collection B of subsets of X is called
a basis for a topology on X if
(i) x ∈ X =⇒ x ∈ B for some B ∈ B,
(ii) B1 , B2 ∈ B and x ∈ B1 ∩ B2 =⇒ there exists B3 ∈ B with x ∈ B3 and
B3 ⊆ B1 ∩ B2 .
The elements of B are called basis elements.
Proposition 3.2. Let B be a basis for a topology on a set X. Let τ be the
collection of all subsets U of X such that if x ∈ U then there exists B ∈ B
with x ∈ B ⊆ U . Then τ is a topology on X (called the topology generated by
the basis B).
Example The collection B of all open balls in a metric space (X, d) is a basis
for the metric topology on X.
Lemma 3.3. Let B be a basis for a topology τ on a non-empty set X. Then a
subset U is open in (X, τ ) if and only if U is a union of basis elements in B.
Lemma 3.4. Let B be a basis for a topology τ on X and let B 0 be a basis for a
topology τ 0 on X. Then τ 0 is finer than τ if and only if for each x ∈ X and each
basis element B ∈ B there exists a basis element B 0 ∈ B 0 with x ∈ B 0 ⊆ B.
Proposition 3.5. Let (X, τ ) be a topological space and let C be a collection of
open sets such that for each x ∈ X and for each open set U with x ∈ U there
exists C ∈ C with x ∈ C ⊆ U . Then C is a basis for the topology τ .
Proposition 3.6. Let (X, τ ) be a topological space and let A be a subset of X.
Then the collection τA = {A ∩ U : U ∈ τ } is a topology on A.
Definition The topology τA is called the subspace topology on A. We call
(A, τA ) a subspace of (X, τ ).
Proposition 3.7. Let (X, τ ) be a topological space and let A be a subset of X.
If B is a basis for a topology on X then BA = {B ∩ A : B ∈ B} is a basis for
a topology on A.
10
Proposition 3.8. Let (X1 , τ1 ), (X2 , τ2 ) . . . , (Xn , τn ) be topological spaces. Let
X = X1 × · · · Xn and let B be the collection of subsets of X of the form
U1 × · · · × Un where for each j, Uj is an open set in (Xj , τj ). Then B is a basis
for a topology on X.
Definition The topology generated by the basis B on X = X1 × · · · Xn is
called the product topology on X.
Proposition 3.9. The product topology on Rn (where we take R with the usual
topology) is the same as the usual topology on Rn .
Definition Let (X, τ ) be a topological space and let x ∈ X. An open set U
which contains x is called a neighbourhood of x.
A sequence in a topological space (X, τ ) is a map s : N → X. As usual we
write (xn )∞
n=1 or (xn ) where xn = s(n) for each n ∈ N.
A sequence (xn ) is said to converge to a point x ∈ X if given any neighbourhood U of x, there exists N ∈ N such that xn ∈ U for all n ≥ N . We call
x a limit of the sequence.
Let A be a subset of X. A point x ∈ X is called a limit point for A if every
neighbourhood of x contains points of A other than x.
Proposition 3.10. Let A be a subset of a topological space (X, τ ) and let
x ∈ X. If there exists a sequence (xn ) in A\{x} which converges to x then x
is a limit point for A.
Proposition 3.11. Let A be a subset of a metric space (X, d). If x is a limit
point for A in the metric topology then there exists a sequence (xn ) in A\{x}
which converges to x.
Definition Let A be a subset of a topological space (X, τ ). The interior of
A, denoted Ao , is the union of all open sets which are contained in A.
The closure of A, denoted Ā, is the intersection of all closed sets which
contain A.
Theorem 3.12. Let A be a subset of a topological space (X, τ ). Then a point
x ∈ X is in Ā if and only if x ∈ A or x is a limit point for A.
Corollary 3.13. Let F be a closed subset of a topological space (X, τ ). If (xn )
is a sequence in F which converges to a point x ∈ X then x ∈ F .
Definition A topological space (X, τ ) is called Hausdorff if for every pair
x, y of distinct points in X there exist disjoint open sets U, V with x ∈ U and
y ∈V.
Example Every metric space (X, d) is a Hausdorff topological space with
respect to the metric topology.
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Proposition 3.14. Let (X, τ ) be a Hausdorff topological space. Then sets
which contain a single point are closed.
Proposition 3.15. Let (X, τ ) be a Hausdorff topological space. Then every
sequence in (X, τ ) converges to at most one limit.
Example Consider R with the Zariski topology. The sequence (n)∞
n=1 converges to every point in R. Thus by the above proposition, the Zariski topology
is not Hausdorff.
Definition Let (X, τ ) and (Y, τ 0 ) be topological spaces. A mapping T : X →
Y is called continuous if the preimage T −1 (V ) is an open set in (X, τ ) for each
V open in (Y, τ 0 ).
Theorem 3.16. Let (X, τ ) and (Y, τ 0 ) be topological spaces and let T : X → Y
be a mapping. The following statements are equivalent:
(i) T is continuous,
(ii) T (Ā) ⊆ T (A) for each subset A of X,
(iii) T −1 (F ) is closed in (X, τ ) for each closed set F in (Y, τ 0 ).
Proposition 3.17. Let (X, τ ), (Y, τ 0 ) and (Z, τ 00 ) be topological spaces. If f :
X → Y and g : Y → Z are continuous then the composition g ◦ f : X → Z is
continuous.
Proposition 3.18. Let (X, τ ) be a topological space and let (A, τA ) be a subspace of (X, τ ).
(i) The inclusion mapping i : A → X, i(x) = x, is continuous.
(ii) If (Y, τ 0 ) is a topological space and f : X → Y is a continuous mapping
then the restriction f |A : A → Y is continuous.
Definition Let (X, τ ) be a topological space. We call (X, τ ) a regular topological space if
(i) sets containing a single point are closed,
(ii) for any closed set F and any point x ∈ X which is not in F , there exist
disjoint open sets U, V with x ∈ U and F ⊆ V .
We call (X, τ ) a normal topological space if
(i) sets containing a single point are closed,
(ii) for any pair of disjoint closed sets F, G, there exist disjoint open sets U, V
with F ⊆ U and G ⊆ V .
12
Proposition 3.19. Every metric space (X, d) is normal with respect to the
metric topology.
Definition Let (X, τ ) and (Y, τ 0 ) be topological spaces. A continuous mapping T : X → Y is called a homeomorphism if T is one-to-one and onto and
the inverse mapping T −1 : Y → X is continuous.
Two topological spaces (X, τ ) and (Y, τ 0 ) are said to be homeomorphic if
there exists a homeomorphism T : X → Y .
Example Every closed interval [a, b] in R is homeomorphic to the interval
[0, 1] (with respect to the subspace topologies induced by the usual topology
on R).
Definition A topological space (X, τ ) is said to be connected if the only subsets of X which are both open and closed are the empty set ∅ and X.
Lemma 3.20. A topological space (X, τ ) is connected if and only if there does
not exist a pair U, V of non-empty disjoint open sets with X = U ∪ V .
Example X = [0, 1] ∪ [2, 3] is not a connected subspace of R since we can
take U = [0, 1] and V = [2, 3].
Proposition 3.21. A topological space (X, τ ) is connected if and only if every
continuous integer-valued function f : X → Z is constant.
Example The closed interval [0, 1] is connected: Let f : [0, 1] → Z be a
continuous integer-valued function. If f (0) 6= f (t) for some t ∈ [0, 1] then
by the Intermediate Value Theorem, for every c between f (0) and f (t) there
exists x with f (x) = c. However this cannot be true since f is integer-valued.
Hence f must be a constant function.
Proposition 3.22. If (X, τ ) and (Y, τ 0 ) are connected topological spaces and
T : X → Y is a continuous mapping then T (X) is a connected subspace of
(Y, τ 0 ).
Proposition 3.23. If (X, τ ) and (Y, τ 0 ) are connected topological spaces then
X × Y is connected (with respect to the product topology).
Proposition 3.24. If A is a connected subspace of a topological space (X, τ )
then the closure of A is also a connected subspace of (X, τ ).
Definition Let (X, τ ) be a topological space and let x ∈ X. The connected
component of x, denoted Cx , is the union of all connected subspaces of X
which contain x.
Proposition 3.25. Let (X, τ ) be a topological space.
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(i) If x ∈ X then Cx is closed and connected.
(ii) If x, y ∈ X then Cx = Cy or Cx ∩ Cy = ∅.
Definition A topological space is called path-connected if for every pair of
points x, y ∈ X there exists a continuous map α : [0, 1] → X with α(0) = x
and α(1) = y. The map α is called a path from x to y.
Proposition 3.26. Every path-connected topological space is connected.
Example Rn is path-connected since if x, y ∈ Rn then we can take the straight
line joining x to y, α : [0, 1] → Rn , α(t) = (1 − t)x + ty.
Definition Let (X, τ ) be a topological space. A collection U of subsets of X
is said to cover X if the union of all elements in U is X.
A collection U of open sets in (X, τ ) which covers X is called an open cover of
X.
If U and V are open covers of X and each element of V is contained in U then
V is called a subcover of U.
A topological space (X, τ ) is called compact if every open cover of X has a
finite subcover.
Theorem 3.27. (Heine-Borel) Every closed interval [a, b] in R is compact.
Lemma 3.28. Let A be a subspace of a topological space (X, τ ). Then A is
compact if and only if for every collection of open sets in (X, τ ) which covers
A there exists a finite subcollection which also covers A.
Proposition 3.29. Every closed subspace A of a compact topological space
(X, τ ) is compact.
Proposition 3.30. Let (X, τ ), (Y, τ 0 ) be topological spaces and let T : X → Y
be a continuous mapping. If A is a compact subspace of X then T (A) is a
compact subspace of Y .
Proposition 3.31. Let (X, τ ) be a Hausdorff topological space. If A is a
compact subspace of X then A is closed.
Theorem 3.32. Let (X, τ ), (Y, τ 0 ) be topological spaces and let T : X → Y be
a continuous bijection. If (X, τ ) is compact and (Y, τ 0 ) is Hausdorff then T is
a homeomorphism.
Lemma 3.33. (The Tube Lemma) Let (X, τ ), (Y, τ 0 ) be topological spaces and
suppose (Y, τ 0 ) is compact. If x0 ∈ X and N is an open set in X × Y which
contains the slice {x0 } × Y then there exists a neighbourhood W of x0 such
that N contains the tube W × Y .
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Theorem 3.34. Let (X1 , τ1 ), . . . , (Xn , τn ) be compact topological spaces. Then
X = X1 × . . . × Xn is a compact topological space (with respect to the product
topology).
Theorem 3.35. A subset K of Rn is compact if and only if K is both closed
and bounded.
Corollary 3.36. (Extreme Value Theorem) Let (X, τ ) be a compact topological
space and let f : X → R be a continuous function. Then there exists u, v ∈ X
such that f (u) ≤ f (x) ≤ f (v) for all x ∈ X.
Lemma 3.37. (Lebesgue Lemma) Let (X, d) be a compact metric space. If
U is an open cover of X then there exists δ > 0 such that each subset A of
X with diameter less than δ is contained in an element of U. (δ is called a
Lebesgue number for U).
Theorem 3.38. (Uniform continuity theorem) Let (X, d), (Y, d0 ) be metric
spaces and let T : X → Y be a continuous mapping. If (X, d) is compact then
T is uniformly continuous.
Definition A topological space (X, τ ) is sequentially compact if every se∞
quence (xn )∞
n=1 in X has a subsequence (xnj )j=1 which converges to a point in
X.
Theorem 3.39. If (X, d) is a compact metric space then (X, d) is sequentially
compact.
Proposition 3.40. If (X, d) is a sequentially compact metric space then (X, d)
is complete.
Definition A metric space (X, d) is called totally bounded if given any > 0
there exists a finite collection of open balls B(x1 , ), . . . , B(xn , ) of radius which covers X.
Theorem 3.41. If (X, d) is a sequentially compact metric space then (X, d)
is totally bounded.
Theorem 3.42. Let (X, d) be a complete metric space. If (X, d) is totally
bounded then it is compact.
In summary, we have shown that for a metric space (X, d) the following
statements are equivalent:
(i) (X, d) is compact,
(ii) (X, d) is sequentially compact,
(iii) (X, d) is complete and totally bounded.
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