Download Proposition S1.32. If { Yα} is a family of topological spaces, each of

Proposition S1.32. If { Yα} is a family of topological spaces, each
of which contains more than one point, and Π Yα. is metrizable,
then each factor Yα is metrizable. (Reference: Dugundji)
Definition S1.28. The diameter of a metric space is defined as:
diam(X, ρ) = supx,y(ρ(x,y)).
If we wish, we can rescale a metric as in Exercise S1.25 so that the
diameter of X is finite, indeed as small as we wish.
Exercise S1.29. Show that if {ρα} is a family of pseudometrics on
X, such that diam(X, ρ) ≤ M < ∞ for all α, then
ρsup(x,y) := supα(ρα(x,y))
is a pseudometric on X.
Exercise S1.30. Show that if {ρn} is a finite or countable family of
pseudometrics on X, then
ρ∑(x,y) := ∑n(ρn(x,y))
is a pseudometric on X, provided that it converges for all x,y ∈ X.
If {ρα} is a family of pseudometrics on spaces {Ya}, then each one
of them also works as a pseudometric on the Cartesian product Π
Yα. To be precise, a pseudometric on Π Yα can be defined by
ρα,#(x,y) := ρα( πα(x), πα(y)). It is straightforward to see that this
satisfies all the conditions of a metric.
Proposition S1.31. A Cartesian product of a countably infinite
family of metric spaces is metrizable. A metric on it is given by
ρsup(x,y) where the metrics {ρn} on the factors have been scaled so
the diameters are all bounded by a finite number M.
This proposition does not guarantee that the metric topology is the
same as the product topology. A necessary and sufficient
condition for this is that lim diam(X n, ρn) = 0.
Theorem S1.24 (Urysohn). Assuming that the topology of X has a
countable base, X is metrizable iff T3. (Reference: Dugundji.)
Suppose we do not want points in some space X to be too far apart,
which they appear to be according to some metric ρ. E.g., we don't
like the fact that points in R2 can be arbitrarily far apart. We can
quantify this situation by defining the diameter of a space as
sup(ρ(x,y): x,y ∈ X), and worrying that the diameter of R2 is
infinite.
Well, we can fix that in two plausible ways. First, we could use
the tangent function to create a homeomorphism between R and I
and using the pull-back trick to make R2 look like I2. This is
somewhat particular to the example of products of R, however. A
more universal method is to the original metric with one defined
by
ρfin(x,y) := ρ(x,y)/(1 + ρ(x,y))
Exercise S1.25: Show that if ρ is a (pseudo)metric on X, then so
is D ρfin for any number D>0, and the diameter of X in the latter
(pseudo)metric is at most D.
Definition S1.26. Two (pseudo)metrics are topologically
equivalent iff their metric topologies are the same.
Exercise S1.27: Show that ρ and D ρfin are topologically
equivalent.
There are two other useful ways to generate new metrics, which
are not generally equivalent to any of the metrics they are built
from. Indeed, it frequently happens that pseudometrics can be
used to produce a true metric by thesee tricks:
Note: We do not always wish to give a Cartesian product the
product topology. The closed unit ball in Hilbert space, given in
Example S1.11 is compact in the weak topology, because it is a
closed subset of the Cartesian product of an infinite number of sets
of the form |xn| ≤ 1, but as we have seen, it is not compact in the
more usual norm topology. It is a difficulty in analysis that the
more usual topology on the this example does not allow it to be
compact.
Example S1.22. To see the difference in a striking way, recall the
sequence of points ek := {δk,n} in Example S1.11. This sequence
converges to zero in the product topology (usually called the weak
Hilbert-space topology), but it does not converge to 0 in norm,
since || ek || = 1 for all k.
More about metrics and pseudometrics.
It is often useful to cook up new metrics, or pseudo metrics, from
old ones with simple tricks. One way to do this is to use a
homeomorphism between two spaces, h: X →Y, where one of the
spaces has a convenient (pseudo)metric ρ, say Y carries the
(pseudo)metric ρ. Then, as is easy to verify, the pull-back
(pseudo)metric ρ∗(x1,x2) := ρ(h(x1),h(x2)) is a (pseudo)metric
defined on X. Structures preserved by homeomorphisms are called
topological invariants.
Definition S1.23. A topological space X is metrizable iff it is
possible to define a metric on X such that its topology is the metric
topology.
According to the preceding paragraph, metrizability is a
topological invariant.
then it is also Hausdorff with respect to any finer topology T1 ⊃ T.
A similar statement applies to the other separation axioms. The
product topology is the roughest topology on Π Yα such that
convergence of a sequence implies convergence of the projection
of the sequence onto any factor Yα. Hence convergence in any
other topology with this property implies convergence in the weak
topology.
Theorem S1.21 (Tychonov). A Cartesian product of topological
spaces, given the product topology, is compact iff each factor is
compact.
Proof that each factor is compact if the product is compact. The
projection map πα is by construction continuous from the product
Π to the factor Xα, and we know that continuous images of
compact spaces are compact.
Proof of the converse, under the added assumption that Π Yα is a
countable product of countably compact spaces. Let {x(k)} be a
sequence in Π Yn. That is, for each k = 1, 2, ..., x(k) is of the form
x(k) = {xn (k)}
If Y1 is compact, there is a subsequence k=k1 beginning with
k1=1, such that x1 (k1) converges as k1 → ∞. We may assume that
If Y2 is compact, there is a subsubsequence k1=k2, beginning with
k2=1 and then k2= the second value of k1, such that both x1(k2) and
x2 (k2) converge. Continuing, we produce sub...subsequences such
that x1(kn) ... xn (kn) converge for all n, where kn = {1, the second
value of k1, the third value of k2, ..., the m+1st value of km...} for
all m < n. In this way we construct a subsequence {x(kj)}, j= 1,2,
..., which converges in the product topology.
an angular coordinate usually called θ or φ, identifying 0 with 2π
in the usual way. R is the usual Euclidean line, and I is the open
unit interval. I and R are homeomorphic (by the map y =
tan(2x/π), so are interchangeable to a topologist.
Examples S1.19.
R2 = R×R. The usual Cartesian coordinate representation
corresponds to this.
I×I is the unit square.
T2 = T1×T1. This is the familiar (2 dimensional) torus. Tn is
referred to as the n-torus. Since S1 = T1, obviously the
notation Sn for the n-sphere (the 2-sphere being ths
usual surface in R3) does not follow the same pattern
for Cartesian products.
1
S ×R is a cylinder. (S1×I would be a cylinder of finite
height, but that would require putting the usual metric
on I, and we haven't spoken of metrics in this section
yet). We can create the 2-sphere by appending a north
and south pole as with the 2-point compactification of
the line: S1×R∪{N}∪{S}, or as a one-point
compactification of the plane, R2∪{N}, using the
stereographic projection to identify the north pole with
the point at infinity.
X
Y := ΠX Y is the set of all functions from X to Y, with no
conditions of regularity at all.
Proposition S1.20. A Cartesian product of Hausdorff spaces
given the product topology is Hausdorff.
For this proposition, we implicitly assumed that the product is
given the product topology. What is we give it some other
topology? If a space is Hausdorff with respect to some topology T,
Exercise S1.12. In a pseudometric space, total boundedness
implies boundedness.
Exercise S1.13. In Rn, total boundedness and boundedness are
equivalent.
Lemma S1.14. Every countably compact pseudometric space is
totally bounded.
Lemma S1.15. A countably compact pseudometric space is
compact.
Finally, with a proof patterned on that of the Heine-Borel
Theorem, we can conclude:
Theorem S1.16. A complete pseudometric space is compact iff it
is closed and totally bounded.
Example S1.17: Hilbert cube. The Hilbert cube is the subset of
the unit ball in the preceding example such that |xn| ≤ 1/2n. (Some
books define it with ≤ 1/n, but then it is not a subset of the unit
ball)
Exercise S1.18. Show that the Hilbert cube is compact in the
topology given by the norm ||{xn}||2 by using the compactness of
closed bounded sets in RN, which is "isomorphic" (identical in
structure) to the set of sequences for which xn=0 when n>N, and
using the triangle inequality to take care of possibly nonzero xn for
n>N.
Supplementum to Cain Chapter 5, on product spaces.
Here are some familiar examples of spaces expressed as Cartesian
products. Here S1 = T1 both denote a circle, for which we can use
radii. The most familiar example is Rn, as the set of points with
rational coordinates is dense.
Definition S1.9. A space with a countable dense subset is called
separable.
Recall that in Rn, a set is compact iff it is closed and bounded.
This is known as the Heine-Borel Theorem. It needs to be
modified for general pseudometric spaces. A key notion is total
boundedness, defined in Cain, above Proposition 6.4. In class
when we defined total boundedness we introduced the term "εnet".
Definition S1.10. An ε-net is a covering of a set by cells (open
balls) of fixed radius ε. A set S is totally bounded iff for every
ε>0, S has a finite ε-net.
Total boundedness is clearly implied by compactness, and it is
easier to check. The important fact about complete metric spaces
is that they are compact iff they are closed and totally bounded.
Boundedness is not equivalent to total boundedness, as shown by
the following example.
Example S1.11: The unit ball in Hilbert space. Let l2 denote the
space of all sequences {xn}of complex numbers such that Σ |xn|2 <
∞}. (Observe that this infinite-dimensional space is a separable
metric space.) The closed unit ball { {xn}: Σ |xn|2 ≤1} is not
compact in the topology given by the norm ||{xn}||2. For the
sequences ek := {δk,n}, where δk,n:= 0 when k≠n and := 1 when k=n,
are a fixed distance apart: ||ej-ek|| = √2 when j≠k. Since the
"points" {ek} cannot be covered by a finite number of balls of
radius < 1/ √2, neither can the unit ball.
T4. Given any pair of closed set A1,2 ⊂ X, there exist disjoint open
sets U1,2 such that A1,2 ⊂ U1,2. A space is normal if it satisfies
axiom T4.
Theorem S1.5 (Urysohn). A space is normal iff for every pair of
closed sets A,B there exists a continuous function fA,B, such that a
∈ A ⇒ fA,B(a) = 0, and b ∈ B ⇒ fA,B(b) = 1. Such a function is
called an Urysohn function.
The main point of compactness, for an analyst, is that sequences in
compact sets have cluster points, and hence convergent
subsequences.
Corollary S1.6 of Cain's Theorem 4.8. Every infinite subset of a
complete compact space has a cluster point. In particular, every
sequence in a compact subset K of a topological space X has a
subsequence that converges in K. This is a version of Cain's
Theorem 6.12.
Euclidean spaces and most of the other spaces we will encounter in
this class have some special structures which make analysis easier.
One of these properties is the fact that they are countably compact.
This allows us to discuss all convergence questions in terms of
countable sets, that is, sequences.
Definition S1.7. A space or set is countably compact iff every
countable open covering has a finite subcovering.
Proposition S1.8. If a topological space has a countable base,
then compactness and countable compactness are equivalent. For
instance, if there is a sequence in a pseudometric space X which is
dense in X, then the topology has a countable base, composed of
open balls centered at the points of the sequence, with rational
Proposition S1.3. If f is continuous and C is connected, then f(C)
is connected.
Corollary S1.4 (the intermediate value theorem). If f is a
continuous function on a connected set C, taking values in R, and
f(a) < f(b), then for all z, f(a) < z < f(b), these exists c ∈ C such
that f(c)= z.
Supplementum to Cain Chapters 4 and 6, on compact spaces
and sequences.
Hausdorff and others investigated the structure of topological
spaces by asking how possible it is to separate points and subsets
by covering them with open sets.
The Axioms of separation. There are four successively stronger
axioms of separation that a topological space might
satisfy:
T1. Given any pair of points x1,x2 ∈ X, there exist open sets U1,2
such that xk ∈ Uk, but x1∉ U2, and x2∉ U1.
T2. Given any pair of points x1,x2 ∈ X, there exist disjoint open
sets U1,2 such that xk ∈ Uk. A space is Hausdorff if it satisfies
axiom T2.
T3. Given any pair of point x ∈ X, and any closed set A ⊂ X with
x∉ A, there exist disjoint open sets U1,2 such that x ∈ U1, A ⊂ U2.
A space is regular if it satisfies axiom T3.
Supplementary lecture notes in Mathematics 6021A, February,
2003.
Supplementum to Cain Chapters 1 and 2.
Often we don't specify a topology with a clearly defined base, but
as the "smallest" topology containing a given collection of subsets.
The set that defines the topology is called a subbase. While a
subbase is not as useful as a base, it will often suffice for our
purposes.
The smallest topology containing a family ∑ of subsets is welldefined because the intersection of topologies is always a topology.
You can see this directly from the definition of a topology.
(Notice, by the way, that it is not automatic that the union of
topologies is a topology.) Since any collection of subsets of X
belongs to the discrete topology on X, it is not vacuous when we
define the smallest topology containing ∑ as the intersection of all
topologies containing ∑.
As an example, the weak topology on X by a family of functions
fα: X → Yα is the one generated by the subbase {fα-1(Uα,β): Uα,β ∈
T(Yα)}.
Supplementum to Cain Chapter 3, on connected spaces.
Example S1.1. If S is a nonempty connected subset of R, then S is
either a single point {x0} or an interval.
Simple exercise S1.2. Show that the intersection of two connected
sets need not be connected.