Infinite Intersection!

Algebra Level 3

( − 1 , 1 ) ∩ ( − 1 2 , 1 2 ) ∩ ( − 1 3 , 1 3 ) ∩ ⋯ = { a } \large(-1,1) \cap (-\frac{1}{2}, \frac{1}{2}) \cap (-\frac{1}{3}, \frac{1}{3})\cap\cdots = \{a\}

Where ( m , n ) = { x ∈ R ∣ m < x < n } (m, n) = \{ x \in \mathbb R | m<x<n \}

For example: ( − 1 , 1 ) = { x ∈ R ∣ − 1 < x < 1 } (-1, 1) = \{ x \in \mathbb R | -1<x<1\}

where R \mathbb R is set of real numbers.

"Enter your answer as the value of a a .

The above gives a classic example of "arbitrary i.e. infinite" intersection of open sets 'need not' be open in R \mathbb R .


The answer is 0.

Akash Patalwanshi by akash patalwanshi , 28, India

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1 solution

( − 1 , 1 ) ∩ ( − 1 2 , 1 2 ) ∩ ( − 1 3 , 1 3 ) ∩ . . . = { 0 } \large(-1,1) ∩ (-\frac{1}{2}, \frac{1}{2}) ∩ (-\frac{1}{3}, \frac{1}{3})∩... = \{0\}

We know that, every finite set is closed in R R , hence { 0 } \{0\} is closed set in R R . Hence this shows that, "arbitrary i.e. infinite" intersection of open sets 'need not' be open in R R . Where R R is set of Real numbers.

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