Page 219 - DMTH503_TOPOLOGY

Unit 25: The Nagata-Smirnov Metrization Theorem

Metrizable: A topological X is metrizable if there exists a metric d on set X that induces the Notes
topology of X.
Neighbourhood: An open set containing x is called a neighbourhood of x.

Product topology: Let X, Y be sets with topologies T and T . We define a topology T on
x y X×Y
X × Y called the product topology by taking as basis all sets of the form U × W where U  T and
X
W  T .
Y
25.4 Review Questions

1. Many spaces have countable bases; but no T space has a locally finite basis unless it is
1
discrete. Prove this fact.
2. Find a non-discrete space that has a countably locally finite basis does not have a countable
basis.
3. A collection  of subsets of X is said to be locally discrete if each point of X has a
neighbourhood that intersects at most one elements of . A collection  is countably
locally discrete if it equals a countable union of locally discrete collections. Prove the
following:
Theorem (Being Metrization Theorem):

A space X is metrizable if and only if it is regular and has a basis that is countably locally
discrete.
4. A topological space is called locally metrizable iff every point is contained in an open set
which is metrizable. Prove that if a normal space has a locally finite covering by metrizable
subsets, then the entire space is metrizable.

25.5 Further Readings

Books Lawson, Terry, Topology: A Geometric Approach, New York, NY: Oxford University
Press, 2003.
Patty. C. Wayne (2009), Foundations of Topology (2nd Edition) Jones and Barlett.
Robert Canover, A First Course in Topology, The Willams and Wilkins Company
1975.

LOVELY PROFESSIONAL UNIVERSITY 213