B. 1
Proof of the Well-ordering Theorem
We follow the pattern outlined in Exercises 2-7 on pp. 72-73 of the text.
Then
h: J-YE.
be well-ordered sets; let
ar;d E
J
Let
Theorem B.1.
the following are equivalent:
of
or a section
E
equals
h(J)
is order preser
K.1
Compactly generated spaces
A space
Definition.
the following condition:
X
is said to be compactly generated if it satisfies
A set
for each compact subspace
C
A
of
is open in
X
if
An C
is open in
C
X.
Said differently, a space is compactly generated
I.1
Locally Euclidean Spaces
The basic objects of study in differential geometry are certain topological
spaces called manifolds.
Oe crucial property that manifolds possess is
that they are locally just like euclidean space.
Formally,
that for each such s
H.1
Tychonoff via well-ordering,
We present a proof of the Tychonoff theorem that uses the well-ordering
theorem rather than Zorn's lemma.
Lemma H.1.
A be a collection of basis elements for the topology
Let
of the product space
If
of
yA
X
It follows the o
G.1
Normality of quotient spaces
For a quotient space, the separation axioms-even the
are difficult to verify.
ausdorff property-
We give here three situations in which the quotient
space is not only Hausdorff, but normal.
Theorem
G.1.
is normal, then
Pro
E.1
Normality of Linear Continua
is normal in the order topology.
X
Every linear continuum
Theorem E.1.
For if
and no smallest element.
form a new ordered set
(,1)
by taking the disjoint union of
Y
X, and
and
X.
to be less than every element of
(,1)
decla
F.l
The separation axioms
We give two examples of spaces that satisfy a given separation
axiom but not the next stronger one.
Te first
is a familiar space,
and the second is not.
Teorem F.1. If
not normal.
Proof.
completel
of R J
is uncountable,
is of cou
A.1
Set Theory and Logic: Fundamental Concepts
(Notes by Dr. J. Santos)
A.1. Primitive Concepts. In mathematics, the notion of a set is a primitive
notion. That is, we admit, as a starting point, the existence of certain objects
(which we call sets), whic
C.1
The Long Line
We follow the outline of Exercise 12 of 24.
Let
denote the set
L
denote the smallest element of
Let o
Lemma C.1.
[o 0 X 0, ckX 0]
interval
S
of
has the order type of
L
0 in 5
S_.
Sppose the lemma holds for all
We show it holds for 3.
.
4
D 1
Countability axioms
We have studied four basic countability properties:
(1)
The first countability axiom.
(2) The second countability axiom,
(3)
(4)
The Lindel6f condition.
tse condition that the space has
a countable dense subset.
We know that condit
J.1
The Pruifer Manifold.
;The so-called Prufer mnanifold is a space that is locally 2-euclidean
and Hausdorff, but not normal.
In discussing it, we follow the outline of
Exercise 6 on p. 317.
Definition.
Le-t
A
be! the following subspace of
gx, y,)
A=
Gi