Page 147 - DMTH503_TOPOLOGY

Unit 15: Local Compactness

Notes
Thus N Y is a closed subset of a compact set N. Hence N Y is compact.
Y is T-closed  T-closure of N Y = U-closure of N Y.

Thus N Y is U-open nbd of y s.t. N Y is compact, showing thereby Y is locally compact at y.

15.2 Summary

 A topological space (X, T) is locally compact if each element x  X has a compact
neighborhood.

 Any open subspace of a locally compact space is a locally compact.
 Every locally compact T -space is a regular space.
2
 Every closed subspace of a locally compact space is locally compact.

15.3 Keywords

Closure: Let (X, T) be a topological space and A  X. The closure of  is defined as the intersection
of all closed sets which contain A and is denoted by the symbol A.
Compact set: Let (X, T) be a topological space and A  X. A is said to be a compact set if every
open covering of A is reducible to finite sub-covering.

Interior point: A point x  A is called an interior point of A if  r  R s.t. S (x)  A.
+
r
Neighborhood: Let  > 0 be any real number. Let x be any point on the real line. Then the set
0
{x  R : | x – x | < } is defined as the -neighborhood of the point x .
0 0
Regular space: A regular space is a topological space in which every nbd of a point contains a
closed neighborhood of the same point.
T -space: A T -space is a topological space (X, T) fulfilling the T -axiom: every two points x,
2 2 2
y  X have disjoint neighborhoods.

15.4 Review Questions

1. Show that the rationals Q are not locally compact.
2. Let X be a locally compact space. If f : X  Y is continuous, does it follow that f(x) is locally
compact? What if f is both continuous and open? Justify your answer.
3. If f : X  X is a homeomorphism of locally compact Hausdorff spaces, show f extends to
1 2
a homeomorphism of their one-point compactifications.
4. Is every open subspace of a locally compact space is locally compact? Give reasons in
support of your answer.
5. Show by means of an example that locally compact space need not be compact.
6. Show that local compactness is a closed hereditary property.

7. X , X are L-compact if and only if X × X is L-compact.
1 2 1 2

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