Answer to Question #171072 in Calculus for hitendra GARG

Question #171072

Prove that the union of two closed sets is a closed set. Give an example to show that union of an infinite number of closed sets need not be a closed set.


1
Expert's answer
2021-03-18T07:07:25-0400

Let A1 and A2 be two closed sets.

(we know that if a set is closed then its complement is open)

therefore A1c and A2c both are open sets.

let S = A1 "\\bigcup" A2

taking complement both side we get: Sc = (A1 "\\bigcup" A2)c

using De Morgan's Law { which states that if A and B are two sets then

(A "\\bigcup" B)c = Ac "\\bigcap" Bc }

therefore using De Morgan's Law we get: Sc = A1c "\\bigcap" A2c

"\\because" the Intersection of a finite collection of open sets is open.

as A1c and A2c both are open sets. therefore A1c "\\bigcap" A2c is an open set.

"\\implies" Sc is an open set

"\\implies" S is closed set

"\\implies" Hence, the union of two closed sets is a closed set


In the second part, we have to give an example to show that the union of an infinite number of closed sets need not be a closed set.

Let An = ["\\frac{1}{n}" , 1-"\\frac{1}{n}" ]

we can clearly see that An is closed for each n

as n "\\to" "\\infty" , "\\frac{1}{n}" "\\to" 0 & (1-"\\frac{1}{n}") "\\to" 1

"\\cup^{\\infty}_{n=1}" An = (0,1) which is an open set.

Hence, the union of an infinite number of closed sets need not be a closed set.







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