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Topology Richa Nandra, Lovely Professional University

Notes Unit 29: Pointwise and Compact Convergence

CONTENTS
Objectives
Introduction
29.1 Pointwise and Compact Convergence

29.1.1 Pointwise Convergence
29.1.2 Compact Convergence
29.1.3 Compactly Generated
29.1.4 Compact-open Topology

29.2 Summary
29.3 Keywords
29.4 Review Questions
29.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Define pointwise convergence and solve related problems;
 Understand the concept of compact convergence and solve problems on it;
 Discuss the compact open topology.

Introduction

X
There are other useful topologies on the spaces Y and (X, Y), in addition to the uniform
topology. We shall consider three of them here: they are called the topology of pointwise
convergence, the topology of compact convergence, and the compact-open topology.

29.1 Pointwise and Compact Convergence

29.1.1 Pointwise Convergence

Definition: Given a point x of the set X and an open set U of the space Y, let
S(x, U) = {f | f  Y and f(x)  U}
X
X
The sets S(x, U) are a sub-basis for topology on Y , which is called the topology of pointwise
convergence (or the point open topology).

Example 1: Consider the space I , where I = [0, 1]. The sequence (f ) of continuous
n
n
functions given by f (x) = x converges in the topology of pointwise convergence to the function f
n
defined by

 0 for 0  x 1
f(x) = 
 1 for x  1

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