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COMPACT, a tutorial by Scott W. Williams |
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4. THE UNIVERSE IN A COMPACT BOX
Given derivation from the study of continuity,
topology concerns itself with things which are close together
while disregarding those which are far apart. Thus, it should
be surprise that we
can bound the metric of a space while keeping the topology by
changing the metric to a new metric, the minimum of 1 and
the old distance between two points. In this metric space with
a fixed closed set F
- the distance d(x,F)
between a point x
and F forms a continuous
function from X[0,1] with value 0
on F.
In general, a reasonable axiom for separating
points and closed sets in a space X is: given a closed set FX and x
X\F there is a continuous g:X
[0,1] such that g(x)=1 and
y
F, g(y)=0. Using the set
, of all continuous f:X
[0,1], as an index, we build
the space P as the
product of
many copies of [0,1]. The space P
is compact [ref] and a copy of the space X "sits" in P.
When a "copy" of X sits in P we say X is embedded in P. The above embedding is denoted by ev, for evaluation, and is defined so that the "f" coordinate of ev(x) in the product is f(x).
This tells us when we can consider our space as part of a compact universe:
4.1. THEOREM. In order for a space X to be contained (or embedded) in a compact space it
is necessary and sufficient that for each pair consisting of a
closed set FX and x
X\F there is a continuous f:X
[0,1] such that f(x)=0 and f(F)=1.
4.2. DEFINITION. The closure of a copy of X in a compact space is called a compactification of X. We think of a compactification as filling in the holes of X because we are allowing certain non-convergent filters in X to converge "outside of X." From this view, I say that the universe is contained in a box - the compact space described before 4.1.
4.3. EXERCISES.
1. The map <cos
t,sin
t> : [0,1]
unit circle establishes that the unit circle is a copy of [0,1] with end points identified; i.e., [0,1] and the unit circle, are both compactifications of (0,1) and the reals. Another compactification of (0,1) is the figure 8, let the sequences converging to 0 and 1 all converge to .5.
2. There is no continuous function f : [0,1]
[-1,1] such that
x
(0,1], f(x)=sin(1/x); i.e., sin(1/x) : (0,1]
[-1,1] has no extension to [0,1]. But by identifying (0,1] with a copy, map x
<x,sin(1/x)>, its graph G in the plane, we do see that (0,1] is embedded into its compactification K=G
({0}
[-1,1]) for which there is a continuous function f : K
[-1,1] which extends sin(1/x), namely, project <x,y>
K to y. Yes, K is a compactification of (0,1].
3. Give an example of a compactification K of the real line, but contained in the plane, to which cosx to [-1,1] can be extended.
4.4. DEFINITION. Prior to Theorem 4.1 above we have described a construction
of what is called the Stone-Cech compactification of ,
denoted by
.
is called
the largest compactification of
because it has the property that each continuous function from
to a compact space
can be extended to a continuous function from
to the same compact space. It is the rule that
is big - for example, when
is the space of positive integers
has more points in it than there
are reals.
(0,1] is also quite large. On
the other hand,
=
!
page 1: the beginnings |
page 2: a special example |
page3: functional separation |
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page4: the universe in a box |
page5: open covers |