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 COMPACT, a tutorial by Scott W. Williams title and preface page 1: the beginnings page 2: a special example page3: functional separation page5: open covers References

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 xX\F there is a continuous g:X[0,1] such that g(x)=1 and yF, 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 xX\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 <cost,sint> : [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, =!

 title and preface page 1: the beginnings page 2: a special example page5: open covers