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Questions › DSE › DSE 2018 › Q48
4 marks, −1Mathematics › Real analysis › Topology of the real line: open, closed and compact setsEasyabout 2 min

DSE 2018 Question 48: Real analysis

Let R\mathbb{R}R be the set of real numbers. A subset EEE of R\mathbb{R}R is said to be open if, for every x∈Ex \in Ex∈E, there exists r>0r > 0r>0 such that (x−r,x+r)(x - r, x + r)(x−r,x+r) is a subset of EEE. Then,

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Let ℝ be the set of real numbers. A subset E of ℝ is said to be open if, for every x ∈ E, there exists r > 0 such that (x - r, x + r) is a subset of E. Then, (A) E₁ ∩ … ∩ Eₙ is open, for every collection of open sets {E₁, …, Eₙ} (B) E₁ ∪ … ∪ Eₙ is open, for every collection of open sets {E₁, …, Eₙ} (C) ∅ is open (D) all of the above are true

Also written as: Let R be the set of real numbers. A subset E of R is said to be open if, for every x in E, there exists r > 0 such that (x - r, x + r) is a subset of E. Then,

Real analysis has come up 38 times in 5 exams since 2005.

What this question tests

This question is on Topology of the real line: open, closed and compact sets, part of Real analysis in Mathematics.

Which sets are open, closed or compact and how unions and intersections behave.

TestStop has 29 past-paper questions on this subtopic.

Practise Real analysisMore on Topology of the real line: open, closed and compact sets

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Question from the DSE 2018 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.