Page 204 - DMTH503_TOPOLOGY

Topology

Notes Now we claim that X is locally compact. It will be so if every point of it has a nbd whose closure
is compact.
*
*
*
*
x  X is fixed and distinct x,   X (Hausdorff)   disjoint open sets A , A in X s.t. x  A and
*
1 2 1
*
  A .
2
But an open set containing  must be of the form
*
A = {}  A
2
where A is an open set in X containing x s.t. its complement is compact.
*
Also   A = A is an open set in X containing x, whose closure is contained in A
*
1 1
*
 A is compact
1
 every point of X has a nbd whose closure is compact
 X is locally compact.
23.1.2 Stone-Cech Compactification
The pair (e, (X)), where X is a Tychonoff space and (X) ( e(x)= ) is called Stone-Cech
compactification of X. e is a map from X into (X).
For each completely regular space X, let us choose, once and for all, a compactification of X
satisfying the extension condition i.e. For a completely regular space X,a compactification Y
of X having the property that every bounded continuous map f : X   extends uniquely to a
continuous map of Y into .
We will denote this compactification of X by (X) and call it the Stone-Cech compactification of
X. It is characterized by the fact that any continuous map f : X  C of X into a compact Hausdorff
space C extends uniquely to a continuous map g : (X)  C.
Theorem 4: Let X be a Tychonoff space, (e, (X)) its stone-cech compactification and suppose
f : X  [0, 1] is continuous. Then there exists a map g : (X)  [0, 1] such that g o e = f, i.e. g is an
extension of f to (X), if we identify X with e(X).
Proof: Letbe the family of all continuous functions from X into [0, 1]. Then (X)[0, 1] we

define g on the entire cube [0, 1] by g() = (f) for [0, 1] .


This is well defined because an element of [0, 1] is a function frominto [0, 1] and can be


evaluated at f since f. Equivalently, g is nothing but the projection f from [0, 1] onto [0, 1],
and hence is continuous. Now if xX then, by definition of the evaluation map, e(x)[0, 1] is

the function e(x) :[0, 1] such that
g o e(x) (h) = h(x) for h.
Now g o e(x) = g(e(x)) = e(x) (f) = f(x)  xX
So g o e = f.

Thus, we extended f not only to (x) but to the entire cube [0, 1] . Its restriction to (X) proves the

theorem.
Theorem 5: A continuous function from a Tychonoff space into a compact Hausdorff space can be
extended continuously over the stone-cech compactification of the domain. Moreover such an
extension is unique.
Proof: Let X be a Tychonoff space, (X) its stone-cech compactification and f : X  Y a map where
Y is a compact Hausdorff space.

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