Download Lecture 13 - Instructure

Lecture 13 outline
LECTURE
METRIZABILITY
• What must we assume about a topology so that it comes from a metric? (A
topological space is said to be metrizable when the topology is equivalent to a metric
topology)
• Properties of metric space:
a) They are Hausdorff.
b) Given any two points, there is a function that equals 0 on one and 1 on the other.
c) Each point has a countable local basis: If p ∈ X, there is a collection {Un}n=1,2,…
with Un+1 ⊂ Un such that if V is an open set containing p, then Un ⊂ V for some n
(and hence for all sufficiently large n.)
d) Sequential continuity is the same as continuous.
e) Sequential compactness is the same as compactness.
• Which of these properties (and perhaps necessarily others?) if obeyed by a space X a
priori imply metrizability?
• Another definition: A space is regular if, for each pair of a point x and a closed set B
not containing x, there exist disjoint open sets, one containing x and the other
containing B.
• Lemma: Metric spaces are regular.
Proof:
a) There exists c > 0 such that dist(p, x) ≥ c if p is in B. If not, then there is a
sequence {pn} ∈ B with dist(pn, x) < 1n . This sequence converges to x, and since
B is closed, the point x is in B which is nonsense.
b) Take O1 to be the ball of radius 12 c centered at x and take O2 to be the
complement of the closure of the ball of radius
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c centered at x. The set O1
contains x and O2 contains B and they are disjoint.
Theorem (Urysohn): Every regular space with a countable basis is metrizable.
Note that Rn has a countable basis. Does Rω = ∏n=1,2,… R with the product topology?
Theorem (Nagata-Smyrnov): A space X is metrizable if and only if it is regular and it
has a basis that is countably locally finite. (This means that there is a basis U that can
be written as a countable union ∪n=1,2,… Un such that Un is locally finite for each n in
the sense that each point p ∈ X is contained in only a countable collection of sets
from Un.)
Are there Hausdorff spaces that are not regular? Extra Credit: Find an example that
is different from the ones on pages 197-198 of the book.
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Another definition: A space is normal if, for each pair of disjoint closed sets A, B,
there are two disjoint open sets, one containing A and the other containing B. Metric
spaces are also normal.
Proof:
a) For each x ∈ A, there is εx so that ball of radius εx is disjoint from B. Let Bx
denote the radius εx/2 ball centered at x.
b) Let OA = ∪x∈A Bx.
c) The closure of OA is disjoint from B. If p is in the closure, then p is has distance
1
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2 εx from some x ∈ A. Then p has distance at least 2 εx from every point of B.
d) For each y ∈ B, there is εy such that the ball of radius εy centered at y is disjoint
from the closure of OA. Let By denote the ball centered at y with half this radius.
e) Set OB = ∪y∈B By. This is disjoint from OA.
URYSOHN METRIZATION THEOREM
• The Urysohn metrization theorem will be proved by embedding the space X as a
subspace of a metric space. The version presented here embeds X in the subspace
[0, 1]ω in Rω.
•
Theorem: Rω = {(x1, x2, …): xi ∈ R} with the product topology is metrizable.
Proof next week.
•
How do we embed X in Rω? This entails finding a set of functions {ƒk: X → R}k=1,2,…
(all continuous) such that F = ∏k ƒk: X → Rω embeds X as a subspace of Rω.
a) First, the functions {ƒk}k=1,2,… must separate points in X. This is to say that given
any two x, y ∈ X, the exists k such that ƒk(x) ≠ ƒk(y).
b) The map F must be an open map onto its image. This is to say that a set O ⊂ X is
open if and only if F(O) ⊂ F(X) is the intersection in Rω of an open set with F(X).
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c) Another way to say this: A basis for the given topology on X are sets of the form
{ƒ1-1(U1) ∩ ···· ∩ ƒn-1(Un): n is a non-negative integer and Uk ⊂ R is open}
The assumption that X is regular will be used to construct, given x ∈ X, a function ƒx
on X that distinguishes x from all others. (It will be zero at x and non-zero elsewhere;
equaling 1 on the complement of an open set around x).
The assumption that X has a countable basis will be used to winnow the collection
{ƒx}x∈X to a countable set of functions that give the desired embedding.
A flow chart for the proof of the Urysohn metrization theorem.
X is regular + countable base −−−−→ X is normal .
X is normal −−−→ Urysohn Lemma
(see below; it constructs functions that distinguish closed sets)
Urysohn lemma + countable base −−−→ Construction of collection {ƒk}k=1,2,…
{ƒk}k=1,2,… −−−−→ The map F: X → Rω .
Prove that F is a homeomorphism to F(X).
Finished!
URYSOHN LEMMA
• Urysohn Lemma: Let X be a normal space and let A, B be disjoint closed subsets.
There is a continuous function ƒ: X → [0, 1] that equals 0 on A and equals 1 on B.