Proof of the Well-ordering Theorem
We follow the pattern outlined in Exercises 2-7 on pp. 72-73 of the text.
be well-ordered sets; let
the following are equivalent:
or a section
is order preser
Compactly generated spaces
the following condition:
is said to be compactly generated if it satisfies
for each compact subspace
is open in
is open in
Said differently, a space is compactly generated
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.
that for each such s
Tychonoff via well-ordering,
We present a proof of the Tychonoff theorem that uses the well-ordering
theorem rather than Zorn's lemma.
A be a collection of basis elements for the topology
of the product space
It follows the o
Normality of quotient spaces
For a quotient space, the separation axioms-even the
are difficult to verify.
We give here three situations in which the quotient
space is not only Hausdorff, but normal.
is normal, then
Normality of Linear Continua
is normal in the order topology.
Every linear continuum
and no smallest element.
form a new ordered set
by taking the disjoint union of
to be less than every element of
The separation axioms
We give two examples of spaces that satisfy a given separation
axiom but not the next stronger one.
is a familiar space,
and the second is not.
Teorem F.1. If
of R J
is of cou
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
The Long Line
We follow the outline of Exercise 12 of 24.
denote the set
denote the smallest element of
[o 0 X 0, ckX 0]
has the order type of
0 in 5
Sppose the lemma holds for all
We show it holds for 3.
We have studied four basic countability properties:
The first countability axiom.
(2) The second countability axiom,
The Lindel6f condition.
tse condition that the space has
a countable dense subset.
We know that condit
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.
be! the following subspace of