Page 253 - DMTH503_TOPOLOGY

Unit 30: Ascoli’s Theorem

30.3 Keywords Notes

Adherent Point: A point xX is called an adherent point of A iff every nhd of x contains at least
one point of A.

Compact Metric Space: If (X, d) be a metric space and AX, then the statement that A is compact,
A is countably compact and A is sequentially compact are equivalent.
Complete Metric Space: A metric space X is said to be complete if every Cauchy sequence of
points in X converges to a point in X.
+
Open Sphere: Let (X,) be a metric space. Let x X and rR . Then set {xX :(x , x) < r} is
o o
defined a open sphere with centre x and radius r.
o
Separable Space: Let X be a topological space and AX, then X is said to be separable if
(i) A = X (ii) A is countable
Totally Bounded: A metric space (X, d) is said to be totally bounded if for every> 0, there is a
finite covering of X by-balls.

30.4 Review Questions

1. Prove that A subset T of  (X) is compact if and only if it is closed, bounded and
equicontinuous.
2. Prove the following:
Theorem: If X is locally compact Hausdorff space, then a subspace T of  (X, R ) in the
n
topology of compact convergence has compact closure if and only if T is pointwise bounded
and equicontinuous under either of the standard metric on R .
n
3. Let (Y, d) be a metric space; let f : XY be a sequence of continuous functions; let f : XY
n
be a function (not necessarily continuous). Suppose f converges to f in the topology of
n
pointwise convergence. Show that if {f } is equicontinuous, then f is continuous and f
n n
converges to f in the topology of compact convergence.
4. Prove the following:
Theorem (Arzela’s theorem, general version). Let X be a Hausdorff space that is-compact;
k
let f be a sequence of functions f : XR . If the collection {f } is pointwise bounded and
n n n
equicontinuous, then the sequence f has a subsequence that converges, in the topology of
n
compact convergence, to a continuous function.
30.5 Further Readings

Books H.F. Cullen, Introduction to General Topology, Boston, M.A.
Stephen Willard, General Topology, (1970).

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