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Exercise 1. 1) Suppose the topological space X is the union of the finite family F₁,..., Fm of closed subsets of X. If f: X→Y is a map from X to the topological space Y, show that f: X→Y is continuous if and only if each restriction f F, F₁Y,i=1,..., m, is continuous. and

Exercise 1. 1) Suppose the topological space X is the union of the finite family F₁,..., Fm of closed subsets of X. If f: X→Y is a map from X to the topological space Y, show that f: X→Y is continuous if and only if each restriction f F, F₁Y,i=1,..., m, is continuous. and

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1.1
Exercise 1. 1) Suppose the topological space X is the union of the finite family F₁,..., Fm of closed subsets of X. If f: X→Y is a map from X to the topological space Y, show that f: X→Y is continuous if and only if each restriction f F₁: F→Y,i= 1,...,m, is continuous. and
Transcribed Image Text:Exercise 1. 1) Suppose the topological space X is the union of the finite family F₁,..., Fm of closed subsets of X. If f: X→Y is a map from X to the topological space Y, show that f: X→Y is continuous if and only if each restriction f F₁: F→Y,i= 1,...,m, is continuous. and
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We use the fact that inverse of a continuous function maps an open (closed) set to an open (closed) set.

 

 

 

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