Summary Metric and Topological Spaces

2. Notation and terminology
• =⇒ : implies some property P • De Morgan’s law part 1:
S T
S\ I Ai = I (S\Ai )
• ⇐⇒ : if and only if • A set containing one element
is called a singleton set • De Morgan’s law part 2:
• a ∈ A: element a is in set A T S
S\ I Ai = I (S\Ai )
• ∪: union
• a ∈
/ A: element a is not in
• A × B: all sets of ordered
set A • ∩: intersection
pairs (a, b) where a ∈ A and
• A ⊆ B: set A is a subset of • B\A: set of elements in B b∈B
or is equal to set B but not in A
• An denoted the set of or-
• {a ∈ A : P (a)}: the subset • s ∈ i∈I Ai ⇐⇒ there ex-
S
dered n-tuples of elements
of elements of A possessing ists i ∈ I such that s ∈ Ai from A


f : X → Y denotes a function f with a domain X that maps to the set Y .
The graph of f is the subset Gf = {(x, y) ∈ X × Y : f (x) = y} of X × Y
f : X → Y is called injective if f (x) = f (x′ ) =⇒ x = x′ and surjective if for every y ∈ Y there exists a
x ∈ X such that f (x) = y.
f is called bijective if it is surjective and injective.
If f : X → Y is a map and A ⊆ X then the restriction of f to A (written f |A), is the map f |A : A → Y
defined by (f |A)(a) = f (a)
If f : X → Y and g : Y → Z are maps then g ◦ f : X → Z defined by (g ◦ f )(x) = g(f (x))
Intervals of R:

• [a, b] = {x ∈ R : a ≤ x ≤ b} • [a, b) = {x ∈ R : a ≤ x < b} • (a, ∞) = {x ∈ R : x ≥ a}

• (a, b) = {x ∈ R : a < x < b} • (−∞, b] = {x ∈ R : x ≤ b}

• (a, b] = {x ∈ R : a < x ≤ b} • (−∞, b) = {x ∈ R : x < b}




1

,3.1 Direct and inverse images
Let f : X → Y be any map, and let A, C be subsets of X, Y respectively.
Definition 3.1 The image f (A) of A under f is the subset Y given by {y ∈ Y : y = f (a) for some a ∈ A}
Definition 3.2 The inverse image f −1 (C) of C under f is the subset of X given by {x ∈ X : f (x) ∈ C}
Proposition 3.6 Suppose that f : X → Y is a map, that A, B ⊂ X and C, D ⊂ Y . Then:

• f (A ∪ B) = f (A) ∪ f (B) • f −1 (C ∪ D) = f −1 (C) ∪ f −1 (D)

• f (A ∩ B) ⊆ f (A) ∩ f (B) • f −1 (C ∩ D) = f −1 (C) ∩ f −1 (D)


Proposition 3.7 Suppose that f : X → Y is a map, and that for each i in some indexing set I we are given
a subset Ai of X and a subset Ci of Y . Then,
!
[ [
−1
f Ci = f −1 (Ci )
!
[ [
f Ai = f (Ai ) i∈I i∈I
i∈I i∈I !
\ \
f −1 Ci = f −1 (Ci )
!
\ \
f Ai ⊆ f (Ai ) i∈I i∈I
i∈I i∈I



Proposition 3.8 Suppose that f : X → Y is a map and B ⊆ X, D ⊆ Y . Then,

f −1 (Y \D) = X\f −1 (D)
f (X)\f (B) ⊆ f (X\B)


Proposition 3.13 Suppose that f : X → Y is a map, B ⊆ Y and for some indexing set I there is a family
S
{Ai : i ∈ I} of subsets of X with X = I Ai . Then,
[
f −1 (B) = (f |Ai )−1 (B)
I



Proposition 3.14 Let X, Y be sets and f : X → Y a map. For any subsets C ⊆ Y we have f (f −1 (C)) =
C ∩ f (X). In particular, f (f −1 (C)) = C if f is onto. For any subset A ⊆ X we have A ⊆ f −1 (f (A))


3.2 Inverse functions
Definition 3.17 A map f : X → Y is said to be invertible if there exists a map g : Y → X such that the
composition f ◦ g is the identity map of Y
Proposition 3.18 a map f : X → Y is invertible if and only if it is bijective
Proposition 3.19 When f is invertible, there is a unique g satisfying definition 3.17. This unique g is called
the inverse of f , written f −1
Proposition 3.20 Suppose that f : X → Y is a one-to-one correspondence of sets X and Y and that
V ⊆ X. Then the inverse image of V under the inverse map f −1 : Y → X equals the image set f (V )


2

, 4.1 Real numbers
4.2 Given a nonempty subset S of R which is bounded above, we call u a least upper bound of S if u is an
upper bound for S and x ≥ u for any upper bound x for S.
4.4 Any nonempty subset of R which is bounded above has a least upper bound
4.4 If a nonempty subset of R is bounded below then it has a greatest lower bound
4.6 The set N of positive integers is not bounded above.
4.7 Between any two distict real numbers x and y there is a rational number
4.9: triangle inequality |x + y| ≤ |x| + |y|
4.10: reverse triangle inequality |x − y| ≥ ||x| − |y||




4.2 Real sequences
4.12 The sequence (sn ) converges to the real number l if given any real number ϵ > 0, there exists an integer
Nϵ such that |sn − l| < ϵ for all n ≥ Nϵ .
4.13 A convergent sequence has a unique limit
4.14 Suppose there is a positive real number K such that given ϵ > 0 there exists N with |sn − l| < kϵ for
all n ≥ N . Then, (sn ) converges to l.
4.15 A sequence (sn ) is said to be monotonic increasing if sn+1 ≥ sn and monotonic decreasing if sn+1 ≤ sn
for all n ∈ N. It is monotonic if it has either of these properties.
4.16 Every bounded monotonic sequence of real numbers converges.
4.17 A sequence (sn ) us a Cauchy sequence if given ϵ > 0 there exists N such that if m, n ≥ N , then
|sm − sn | < ϵ
4.18: Cauchy’s convergence criterion A sequence (sn ) of real numbers converges if and only if it is a
Cauchy sequence.
4.19 Every bounded sequence of real numbers has atleast one convergent subsequence.
4.20 Suppose that (sn ) → s and (tn ) → t. Then, (sn + tn ) → s + t, (sn tn ) → st and, provided t ̸= 0,
1/(tn ) = 1/t.




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