Page 200 - DMTH503_TOPOLOGY

Topology

Notes 22.3 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 finite sub-covering.
Maximal: Let (A, ) be a partially ordered set. An element a A is called a maximal element of
A if  no element in A which strictly dominates a, i.e.
x a for every comparable element x  A.
Projection Mappings: The mappings

 x ; X Y  X s.t.  x (x,y) x (x,y) X Y






 y ; X Y  Y s.t.  y (x,y)  y (x,y) X Y


are called projection maps of X × Y onto X and Y space respectively.
Tychonoff Space: It is a completely regular space which is also a T -space i.e. T = [CR] + T .
1 3 1 1
2
Upper bound: Let A R be any given set. A real number b is called an upper bound for the set A
if.
x  b  x  A.
22.4 Review Questions

1. Let X be a space. Let D be a collection of subsets of X that is maximal with respect to the
finite intersection property.
(a) Show that x D for every D  if any only if every neighbourhood of x belongs to
. Which implication uses maximality of ?
(b) Let D . Show that if A D, then A .
(c) Show that if X satisfies the T axion, there is at most one point belonging to  D.
1 D
2. A collection  of subsets of X has the countable intersection property if every countable
intersection of elements of  is non-empty. Show that X is a Lindelöf space if any only if
for every collection  of subsets of X having the countable intersection property,
 A
A
is non-empty.

22.5 Further Readings

Books Bimmons, Introduction to Topology and Modern Analysis.

Nicolas Bourbaki, Elements of Mathematics.

Online links www.planetmath.org
www.jstor.org

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