Page 206 - DMTH503_TOPOLOGY

Topology

Notes 23.3 Keywords

Connected Spaces: A space X is connected if and only if the only subsets of X that are both open
and closed in X are the empty set and X itself.

Hausdorff Space: It is a topological space in which each pair of distinct points can be separated
by disjoint neighbourhoods.
Homeomorphism: A map f : (X, T)(Y, ) is said to be homeomorphism if:

(i) f is one-one onto.
–1
(ii) f and f are continuous.
23.4 Review Questions

1. Let (X, T) be a Tychonoff space and (X, T) its stone-cech compactification. Prove that (X, T)
is connected if and only if (X, T) is connected.
[Hint: Firstly verify that providing (X, T) has at least 2 points it is connected if and only if
there does not exist a continuous map of (X, T) onto the discrete space {0, 1}.]

2. Let (X, T) be a Tychonoff space and (X, T) its stone-cech compactification. If (A, T ) is a
1
subspace of (X, T) and A  X, prove that (X, T) is also the stone-cech compactification of
(A, T ).
1
3. Let (X, T) be a dense subspace of a compact Hausdorff space (Z, T ). If every continuous
1
mapping of (X, T) into [0, 1] can be extended to a continuous mapping of (Z, T ) into [0, 1],
1
prove that (Z, T ) is the Stone-Cech compactification of (X, T).
1
4. Let Y be an arbitrary compactification of X; let (X) be the Stone-Cech compactification.
Show that there is a continuous surjective closed map g : (X)  Y that equals the identity
on X.
5. Under what conditions does a metrizable space have a metrizable compactification?

23.5 Further Readings

Books S. Lang, Algebra (Second Edition), Addison-Wesley, Menlo Park, California 1984.
S. Willard, General Topology, MA : Addison-Wesley.

Online links www.planetmath.org

www.jstor.org

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