Page 142 - DMTH503_TOPOLOGY

Topology

Notes So that D({x : n  N})  D(A).
n
But x  D{x : n  N} and hence x  D(A), i.e., A has a limit point x  X.
0 n 0 0
Proof II: Conversely, suppose that the metric space (X, d) has Bolzano Weierstrass property.
To prove that X is sequentially compact.

By the assumption of Bolzano Weierstrass property, every infinite subset of X has a limit point
in X. Let x  be an asbitrary sequence in X.
n
Case (i): If the sequence x  has an element x which is infinitely repeated, then it has a constant
n
subsequence x, x, …, x, … which certainly converges to x.
Case (ii): If the sequence x  has infinitely many distinct points then by assumption, the set
n
{x : x  N} has a limit point, say x  X. Consequently x is a limit of the sequence x : n  N
n 0 0 n
with infinitely many distinct points so that this sequence contains a subsequence x : n  N}
in
which also converges to X .
0
 In either case, we have shown that every sequence in X contains a convergent subsequence so
that X is sequentially compact.
Hence the result.

14.2 Summary

 A space X is said to be limit point compact if every infinite subset of X has a limit point.
 Compactness implies limit point compactness, but not conversely.
 A topological space X is said to be sequentially compact if every sequence of points of X
has a convergent subsequence.

14.3 Keywords

BWP: A topological space (X, T) is said to have Bolzano Weierstrass Property denoted by BWP
if every infinite subset has a limit point.
Compact Space: A space X is said to be compact if every open covering A of X contains a finite
subcollection that also covers X.
Lebesgue Covering Lemma: Every open covering of a sequentially compact space has a Lebesgue
number.
Lebesgue Number: Let {G : i } be an open cover for a metric space (X, d), a real number  > 0
i
is called a Lebesgue number for the cover if any A  X s.t. d(A) <  A  G for at least one index
i0
i .
0
Metrizable: Any topological space (X, T), if it is possible to find a metric on  on X which induces
the topology T i.e. the open sets determined by the metric are precisely the members of , then
X is said to the metrizable.
Open Cover: Let (X, T) be a topological space and A X. Let G denote a family of subsets of X.
G is called a cover of A if A U {G : G G}. If every member of G is a open set, then the cover G
is called an open cover.

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