Page 248 - DMTH503_TOPOLOGY

Topology

Notes 29.3 Keywords

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 fine sub-covering.

Locally compact: Let (X, T) be a topological space and let x  X be arbitrary. Then X is said to be
locally compact at x if the closure of any neighbourhood of x is compact.
Subbase: Let (X, T) be a topological space. Let S  T s.t. S  

S is said to be a sub-base or open sub-base for the topology T on X if finite intersections of the
members of S form a base for the topology T on X i.e. the unions of the members of S give all the
members of T. The elements of S are referred to as sub-basic open sets.

29.4 Review Questions

1. Show that the set ( , ) of founded functions f :  is closed in in the uniform
topology, but not in the topology of compact convergence.
2. Consider the sequence of functions
f : (–1, 1)  , defined by
n
n
f (x) =  Kx k
n
k 1

(a) Show that (f ) converges in the topology of compact convergence, conclude that the
n
limit function is continuous.
(b) Show that (f ) does not converge in the uniform topology.
n
3. Show that in the compact-open topology, (X, Y) is Hausdorff if Y is Hausdorff, and
regular if Y is regular.

[Hint: If U  V, then S(C,U)  S(U, V)]

4. Show that if Y is locally compact Hausdorff then composition of maps
(X, Y) × (Y, Z)  (X, Z)
is continuous, provided the compact open topology is used throughout.

29.5 Further Readings

Books J.L. Kelly, General Topology, Van Nostrand, Reinhold Co., New York.

J. Dugundji, Topology, Prentice Hall of India, New Delhi, 1975.

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