Download 1 Bases 2 Linearly Ordered Spaces

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Bases
The weight of a topological space X, w (X), is the smallest size of a basis for
X.
1. Show that if B is a basis for a topological space X then there is B ′ ⊆ B
such that B ′ is a basis for X and |B ′ | = w (X). (If you wish, you may do
this in the special case where the weight of X is countable.
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Linearly Ordered Spaces
A linear (or total) order on a set is a transitive, reflexive relation ≤ such that
x ≤ y and y ≤ x implies x = y.
If ≤ is a linear order on a set X and x ∈ X, we write (−∞, x) = {y ∈ X : y ≤ x, y 6= x}
and (x, ∞) = {y ∈ X : x ≤ y, x 6= y}. We may use other intervals in the natural
way. Similarly, inf, sup, min, max have their usual order-theoretic meanings.
The order topology on X is generated by the subbase {(−∞, x) : x ∈ X} ∪
{(x, ∞) : x ∈ X}. A topological space whose topology is the order topology
for some linear order on the underlying set is called a LOTS (linearly ordered
topological space).
1. Separation Axioms in a LOTS
(a) Verify that every LOTS is Hausdorff.
(b) Show that if x ∈ X, A ⊆ X then x ∈ A if and only if there is B ⊆ A
such that x = inf B or x = sup B.
(c) Show that every LOTS is regular.
(d) Show that every LOTS is normal.
[Hint (slightly incorrect, but should give the idea): Show that
a LOTS has a halving operator as defined on Sheet 1. To that end,
well-order the points of the topological space and define H(x, U ) =
(xl , xr ) where xl , xr are the least (according to the well-order) points
with (xl , xr ) ⊆ U .]
(e) Deduce that every subspace of a LOTS is normal, i.e. that order
topologies are hereditarily normal.
2. Examples (Optional except for the last one)
(a) Observe that the following spaces are LOTS (i.e. briefly write down
an order inducing the natural topology): closed/open unit interval,
rationals, irrationals, Cantor Set, a convergent sequence, ordinals.
The double arrow space is [0, 1] × {0, 1} with the lexicographic order, i.e.
(x, s) ≤ (y, t) if and only if x ≤ y and x = y =⇒ s ≤ t. We often write
− for 0 and + for 1.
(b) Draw a picture of the double arrow space.
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(c) Write down some non-trivial clopen (closed and open) subsets of the
double arrow space.
(d) Which of the above mentioned LOTS are subspaces of the double
arrow space and why (the proofs to some of these might be quite
hard)?
(e) Show that the Sorgenfrey Line is a subspace of the double arrow
space.
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Normality, Tietze’s Theorem, Jone’s Lemma
Note that the results in this section are not examinable as bookwork, but that
the techniques presented are used all over topology.
An Fσ set in a topological space is a countable union of closed sets.
Two subsets A, B of a topological space are separated if and only if A ∩ B =
∅ = A ∩ B.
As always, X is a fixed topological space. Throughout this question assume
that X is normal, C is a closed subset of X and f : C → [0, 1] is a continuous
function.
1. Show that separated Fσ sets in X can be separated by open sets, i.e. if
A, B are separated Fσ sets then there are disjoint open U ⊇ A and V ⊇ B.
2. Show that for every r ∈ (0, 1), f −1 ([0, r)) and f −1 ((r, 1]) are separated
Fσ sets.
3. Tietze’s Extension Theorem: Show that there are open sets Uq , q ∈
(0, 1) ∩ Q such that for r < s ∈ (0, 1) ∩ Q, Ur ⊆ Us , f −1 ([0, r)) ⊆ Ur and
f −1 ((r, 1]) ⊆ X \ Ur . Hence deduce that there is a continuous function
F : X → [0, 1] which extends f , i.e. such that F |C = f . [You may wish to
follow the proof of Urysohn’s Lemma.]
4. Deduce Urysohn’s Lemma from Tietze’s Extension Theorem.
5. Optional: Assume Urysohn’s Lemma, but not Tietze’s Theorem. By
considering A = f −1 ([−1, −1/3]) and B = f −1 ([1/3, 1]) or otherwise
find a continuous function g : X → [−1/3, 1/3] such that (f − g) (C) ⊆
[−2/3, 2/3]. Using induction and the M -test, prove Tietze’s Theorem.
6. Jone’s Lemma (requires a little cardinal arithmetic): How many continuous [0, 1]-valued functions on X are there (at most) if X contains a
countable dense subset? How many continuous [0, 1]-valued functions on
X are there at least if X contains a closed discrete subspace of size continuum? Deduce that if X contains a countable dense subset then it cannot
contain a closed discrete subspace of size continuum.
7. By considering the anti-diagonal in the Sorgenfrey Plane (the square of
the Sorgenfrey Line) or otherwise, show that the Sorgenfrey plane is not
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normal and deduce that the Sorgenfrey Line cannot have a countable
basis. Note that the Sorgenfrey Line is first countable and normal but not
metrizable.
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