Page 220 - DMTH503_TOPOLOGY

Topology Richa Nandra, Lovely Professional University

Notes Unit 26: The Smirnov Metrization Theorem

CONTENTS
Objectives
Introduction

26.1 Locally Metrizable Space
26.2 Summary
26.3 Keywords

26.4 Review Questions
26.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Understand the locally metrizable space;

 Explain the Smirnov Metrization theorem.
Introduction

The Nagata-Smirnov metrization theorem gives one set of necessary and sufficient conditions
for metrizability of a space. In this, unit we prove a theorem that gives another such set of
conditions. It is a corollary of the Nagata-Smirnov theorem and was first proved by Smirnov.
This unit starts with the definitions of paracompact and locally metrizable space. After explaining
these terms, proof of “The Smirnov Metrization Theorem” is given.
26.1 Locally Metrizable Space

A space X is locally metrizable if every point x of X has a neighborhood that is metrizable in
the subspace topology.

The Smirnov Metrization Theorem

Statement: A space X is metrizable if and only if it is a paracompact Hausdorff space that is
locally metrizable.
Proof: Suppose that X is metrizable.
Then X is locally metrizable; it is also paracompact. [Every metrizable space is paracompact].

Conversely, suppose that X is a paracompact Hausdorff space that is locally metrizable.
We shall show that X has a basis that is countably locally finite. Since X is regular, it will then
follow from the Nagata – Smirnov theorem that X is metrizable.

Cover X by open sets that are metrizable; then choose a locally finite open refinement  of this
covering that covers X. Each element C of  is metrizable, let the function d : CCR be a
C
metric that gives the topology of C. Given xC, let B (x,) denote the set of all points y of C
C
such that d (x, y) <. Being open in C, the set B (x,) is also open in X.
C C

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