GATE MA concept:metric-spaces PYQs & Answers

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concept:metric-spaces

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Questions 1–6 of 6. Imported answers require independent review unless marked reviewed.

Topic practice 6 questionsPage 1 of 1

For an integer

, let

, where

. Let

. Consider the metric space

, where

Which of the following statement(s) is/are true?

Your answerSelect all correct options. You can choose more than one.

is an equi-continuous family of continuous functions

is closed in

is bounded in

is compact in

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Let

. Consider the metric space

, where

Let

for all

and

Let

be defined by

and

for all

. Consider the following statements:

is in the interior of

. Which of the following statements is correct?

Your answerChoose one option.

is TRUE and

is FALSE

is FALSE and

is TRUE

Both

and

are FALSE

Both

and

are TRUE

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Let

denote the power set of

, equipped with the metric

, where

and

denote the characteristic functions of the subsets

and

, respectively, of

. The set

in the metric space

Your answerChoose one option.

bounded but not totally bounded

totally bounded but not compact

compact

not bounded

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Suppose that

and

are metric spaces with metrics induced by the Euclidean metric of

. Let

and

be the open unit balls around

and

, respectively. Consider the following statements: I: The closure of

is compact. II: The closure of

is compact. Then

Your answerChoose one option.

both I and II are true

I is true but II is false

I is false but II is true

both I and II are false

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Let

and only finitely many

's are non-zero

and

be a metric on

defined by

(

is the set of all real numbers and

is the set of all natural numbers)

Consider the following statements:

is a complete metric space.

: The set

is compact, where

is the zero element of

Which of the above statements is/are TRUE?

Your answerChoose one option.

Both

and

only

Neither

nor

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Let

and

be two sequences in a complete metric space

such that

and

for all

. Then

Your answerChoose one option.

both

and

converge

converges but

need NOT converge

converges but

need NOT converge

neither

nor

converges

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GATE MA concept:metric-spaces PYQs

How to practice concept:metric-spaces

Question 55

Question 20

Question 30

Question 35

Question 7

Question 31