GATE MA topic:topology PYQs & Answers — Page 2

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topic:topology

Questions 11–20 of 46. Imported answers require independent review unless marked reviewed.

Topic practice 46 questionsPage 2 of 5

Let

be an uncountable set. Let the topology on

be defined by declaring a subset

to be open if

is either empty or finite or countable, and the empty set to be open. Then, which of the following is/are TRUE?

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

Every compact subset of

is closed

Every closed subset of

is compact

(singleton subsets are closed) but not

(Hausdorff)

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Let

and

be two metric spaces with

and

If the open ball centred at

with radius

is equal to the open ball centred at

with radius

, then the value of

is _____ (in integer)

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Let

be a topological space and

. Given a subset

, let

, and

denote the interior, boundary, and closure, respectively, of the set

. Which one of the following is NOT necessarily true?

Your answerChoose one option.

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Consider the following statements.

I. Every compact Hausdorff space is normal.

II. Every metric space is normal.

Which one of the following is correct?

Your answerChoose one option.

Both I and II are TRUE

I is TRUE and II is FALSE

I is FALSE and II is TRUE

Both I and II are FALSE

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Consider the topology on

with basis

, where

Consider the following statements.

is both open and closed for each

with

II. The only connected set containing

Which one of the following is correct?

Your answerChoose one option.

Both I and II are TRUE

I is TRUE and II is FALSE

I is FALSE and II is TRUE

Both I and II are FALSE

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Let

be the space

with the quotient topology induced from the usual topology on

. Consider the following statements.

is compact.

II.

is connected for any

Which one of the following is correct?

Your answerChoose one option.

Both I and II are TRUE

I is TRUE and II is FALSE

I is FALSE and II is TRUE

Both I and II are FALSE

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Let

be a basis for a topology

. Which of the following is/are correct?

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

Every

with

is an open set in

Every

with

is compact in

is a first-countable space

is a second-countable space

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Let

be a topological space with the usual topology and

denote the set of rational numbers. Define the subspaces

and

as follows:

;

. Which of the following statements is correct?

Your answerChoose one option.

is homeomorphic to

is NOT homeomorphic to

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Consider

with the usual Euclidean metric. Let

and

Consider the following statements:

is a connected subset of

. Then

Your answerChoose one option.

both

and

are TRUE

is FALSE and

is TRUE

is TRUE and

is FALSE

both

and

are FALSE

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Let

and

be two topological spaces. A continuous map

is said to be proper if

is compact in

for every compact subset

, where

is defined by

. Consider

with the usual topology. If

has the subspace topology induced from

and

has the product topology, then which of the following maps is proper?

Your answerChoose one option.

defined by

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GATE MA topic:topology PYQs

Question 58

Question 65

Question 13

Question 17

Question 18

Question 41

Question 50

Question 12

Question 18

Question 24