GATE MA topic:topology PYQs & Answers — Page 3

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topic:topology

Questions 21–30 of 46. Imported answers require independent review unless marked reviewed.

Topic practice 46 questionsPage 3 of 5

Let

be a topological space, where the topology

is defined as

Which of the following statements is/are correct?

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

is first countable

is Hausdorff

is separable

The closure of

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Consider

with the usual metric. Let

and

. Let

and

. Then, which of the following statements is TRUE?

Your answerChoose one option.

and

are connected

Neither

nor

is connected

is connected and

is not connected

is not connected and

is connected

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Consider

, where

is the subspace topology induced by the Euclidean topology on

, and let

be any topology on

. Consider the following statements:

P: If

is a proper subset of

, then

is not compact. Q: If

is a proper subset of

, then

is not Hausdorff.

Then

Your answerChoose one option.

P is TRUE and Q is FALSE

Both P and Q are TRUE

Both P and Q are FALSE

P is FALSE and Q is TRUE

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Let

be the quotient map, arising from the characteristic function on

, where

is the subspace topology induced by the Euclidean topology on

. Which of the following statements is TRUE?

Your answerChoose one option.

is an open map but not a closed map

is a closed map but not an open map

is a closed map as well as an open map

is neither an open map nor a closed map

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Set

for each

. Define

. Endow

with the product topology, where the topology on each

is the Euclidean topology. Consider the set

with the subspace topology induced from

. Which of the following statements is TRUE?

Your answerChoose one option.

is open in

is locally compact

is dense in

is disconnected

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Let

be the metric space with

. Let

. Then, the set

Your answerChoose one option.

closed but not compact

not closed but compact

compact

neither closed nor compact

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Let

, where

is the smallest topology on

in which all the singleton sets are closed. Then, which of the following statements are TRUE?

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

is compact in

is not first countable

is second countable

is first countable

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Consider

, where

is the topology generated by sets of the form

, for

and

. Then, which of the following statements are TRUE?

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

is connected

Each

is a closed subset of

is Hausdorff

is metrizable

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Consider the following statements: P: Every compact metrizable topological space is separable. Q: Every Hausdorff topology on a finite set is metrizable. Then

Your answerChoose one option.

both P and Q are TRUE

P is TRUE and Q is FALSE

P is FALSE and Q is TRUE

both P and Q are FALSE

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Consider the following topologies on the set

of all real numbers:

. Then the closure of the set

Your answerChoose one option.

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GATE MA topic:topology PYQs

Question 44

Question 17

Question 22

Question 23

Question 24

Question 41

Question 53

Question 54

Question 13

Question 14