MATH 553
A. Maps and Metric Spaces
2017 January 4
Hopefully most of the following ideas are familiar. Definitions for words in italics are given in these notes,
definitions for words in bold face should be looked up elsewhere if they are unfamiliar to you.
1. Maps
Let X, Y be sets, and let U ⊂ X. A map, or function,
f :U →Y
is a rule that assigns to every element x in U a unique element y = f (x) in Y . The subset U is called the
domain of definition of f ,
domain(f ) = U.
If the precise description of U is not important, to simplify notation we will often write
f : X → Y,
even if the domain of definition of f is not all of X.
If U0 ⊂ U , its image is
f (U0 ) = {y ∈ Y : there exists x ∈ U0 such that y = f (x)}.
The range of f is the image of its domain,
range(f ) = f (U ).
If V ⊂ Y , its preimage is
f −1 (V ) = {x ∈ U : f (x) ∈ V }.
A map f is one-to-one or is an injection if for every y ∈ range(f ) the preimage f −1 ({y}) consists of
precisely one point x ∈ domain(f ). In this case the inverse map or function
f −1 : range(f ) → domain(f )
is defined, f −1 (f (x)) = x for all x ∈ domain(f ), and f (f −1 (y)) = y for all y ∈ range(f ). A map f is
invertible on X if it is one-to-one and f −1 (Y ) = X
A map f is onto or is an surjection if range(f ) = Y . If a function is both one-to-one and onto, then it
is called a one-to-one correspondence or a bijection. To simplify notation, for a one-to-one map f we will
often write for its inverse map
f −1 : Y → X,
even if the range of f is not all of Y or the domain of f is not all of X.
2. Metric spaces
A metric space is a set X (whose elements are called points) together with a function
ρ : X × X → R,
called a metric, or distance, having the following properties:
i) ρ(x, y) = ρ(y, x) for all x, y ∈ X;
ii) ρ(x, y) ≥ 0 for all x, y ∈ X, and equality holds iff x = y;
iii) ρ(x, y) ≤ ρ(x, z) + ρ(z, y) for all x, y, z ∈ X.
A well known example of a metric space is the set of all real numbers R, with the metric ρ(x, y) = |x − y|
(here | · | denotes the absolute value). Another example of a metric space is Rn . If x = (x1 , · · · , xn )T ,
y = (y1 , · · · , yn )T ∈ Rn , then
p
ρ(x, y) = kx − yk = (x1 − y1 )2 + · · · + (xn − yn )2
(here k · k denotes the Euclidean norm) defines a metric.
Subsets of metric spaces can be bounded.
Every convergent sequence is a Cauchy sequence. A metric space is complete if every Cauchy
sequence is a convergent sequence. For example, the set of all real numbers R with the metric above is a
complete metric space, while the set of all rational numbers Q with the same metric is also a metric space
but it is not a complete metric space.
We have limit points and isolated points of subsets.
We also have open subsets, the interior of an arbitrary subset, and a neighbourhood of a point x ∈ X
is a subset that contains (or is) an open subset that contains the point x. Sometimes we specify open
neighbourhoods.
We define closed subsets as the complements of open subsets or equivalently, as subsets that contain
all of their limit points. The closure of an arbitrary subset is the union of the subset and all of its limit
points. A subset is dense in X if its closure is X, and a subset is compact if every open cover (a collection
of open subsets whose union contains the given subset) of the subset contains a finite subcover.
Some useful facts: (a) a closed subset of a complete metric space is itself a complete metric space, (b)
every compact subset of a metric space is closed, and (c) every compact metric space is complete.
Two subsets U and V are separated if no point of U lies in the closure of V and no point of V lies in
the closure of U . A subset is connected if it is not a union of two nonempty separated subsets.
If X and Y are metric spaces, we will say a function f is locally defined at a point x0 ∈ X if the domain
of f is (or can be taken as) a neighbourhood of x0 . As noted above, in this case we often write f : X → Y
even if the domain of f is not actually all of X, with the understanding that we are interested in f (x)
only for x belonging the neighbourhood of x0 . A function f : X → Y is bounded if its range is a bounded
subset of Y . We also have the important notion of a continuous function.
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