Page 202 - DMTH503_TOPOLOGY

Topology

Notes
Then Y is a compact Hausdorff space and X = Y ; therefore, Y is a compactification of X .
0 0 0 0 0
We now construct a space Y containing X such that the pair (X, Y) is homeomorphic to the pair
(X , Y ). Let us choose a set A disjoint from X that is in bijective correspondence with set Y – X
0 0 0 0
under map K : AY – X .
0 0
Define Y = X A, and define a bijective correspondence H : YY by the rule
0
H(x) = h(x) for xX,
H(a) = k(a) for aA.
Then topologize Y by declaring to be open in Y if and only if H( ) is open in Y . The map H is
0
automatically a homeomorphism; and the space X is a subspace of Y because H equals the
homeomorphism ‘h’ when restricted to the subspace X of Y. By expanding the range of H, we
obtain the required imbedding of Y into Z.
Now suppose Y is a compactification of X and that H : Y Z is an imbedding that is an
i i i
extension of h, for i = 1, 2. Now H maps X onto h(X) = X . Because H is continuous, it must map
i 0 i
Y into X ; because H (Y ) contains X and is closed (being compact), it contains X . Hence, H (Y ) =
i 0 i i 0 0 i i
-
1
X and H o H defines a homeomorphism of Y with Y that equals the identity on X.
0
2
1
2
1
Theorem 1: The collection of all compactifications of a topological space is partially ordered by .
If (f, Y) and (g, Z) are Hausdorff compactifications of a space and (f, Y)  (g, Z)  (f, Y), then (f, Y)
and (g, Z) are topologically equivalent.
Proof: If (f, Y)  (g, Z)  (h, U), where these are compactification of a space X, then there are
continuous functions j on Y to Z and K on Z to U such that g = j o f and h = k o g and hence
h = k o j o f and (f, Y)  (h, U). Consequently  partially orders the collection of all compactifications
of X. If (f, Y) and (g, Z) are Hausdorff compactifications each of which follows the other relative
to the ordering , then both f o g and g o f have continuous extensions j and k to all of Z and
–1
–1
Y respectively.
Since k o j is the identity map on the dense subset g [X] of Z and Z is Hausdorff k o j is the identity
map of Z onto itself and similarly j o k is the identity map of Y onto Y. Consequently (f, Y) and
(g, Z) are topologically equivalent.

23.1.1 One Point Compactification

Definition: Let X be a locally compact Hausdorff space.
Take some objects outside X, denoted by the symbol  for convenience and adjoin it to X,
forming the set
Y = X  {}.
Define topology  on Y as follows:
(i) G   if T

(ii) Y – C   if C is a compact subset of X.
The space Y is called one point compactification of X.
Theorem 2: Let X be a locally compact Hausdorff space which is not compact. Let Y be one point
compactification of X. Then Y is compact Hausdorff space : X is a subspace of Y : the set Y – X
consists of a single point and X = Y.

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