GATE MA Topology PYQs & Answers — Page 4

Questions 31–40 of 46. Imported answers require independent review unless marked reviewed.

Topic practice 46 questionsPage 4 of 5

Consider the following topologies on the set

of all real numbers.

is the upper limit topology having all sets

as basis.

is the standard topology having all sets

as basis. Then

Your answerChoose one option.

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Let

denote the set of all real numbers. Consider the following topological spaces.

, where

is the upper limit topology having all sets

as basis.

, where

. Then

Your answerChoose one option.

both

and

are connected

is connected and

is NOT connected

is NOT connected and

is connected

neither

nor

is connected

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Suppose that

and

are topologies on

induced by metrics

and

, respectively, such that

. Then which of the following statements is TRUE?

Your answerChoose one option.

If a sequence converges in

then it converges in

If a sequence converges in

then it converges in

Every open ball in

is an open ball in

The map

from

is continuous

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Suppose that

Then, with respect to the Euclidean metric on

Your answerChoose one option.

both

and

are disconnected

is disconnected but

is connected

is connected but

is disconnected

both

and

are connected

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For a subset

of a topological space, let

and

denote the interior and closure of

, respectively. Then which of the following statements is TRUE?

Your answerChoose one option.

is open, then

If the boundary of

is empty, then

is open

If the boundary of

is empty, then

is not closed

is a proper subset of the boundary of

, then

is open

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Suppose

and

are the smallest topologies on

containing

and

, respectively, where

. Then

Your answerChoose one option.

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Let

be the co-countable topology on

(the set of real numbers) and

be the co-finite topology on

Consider the following statements:

I. In

, the sequence

converges to 0.

II. In

, the sequence

converges to 0.

III. In

, there is no sequence of rational numbers which converges to

IV. In

, there is no sequence of rational numbers which converges to

Which of the above statements are TRUE?

Your answerChoose one option.

I and II only

II and III only

III and IV only

I and IV only

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Consider the ordered square

, the set

with the dictionary order topology. Let the general element of

be denoted by

, where

. Then the closure of the subset

Your answerChoose one option.

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Let

and

be two topologies defined on

(the set of all natural numbers), where

is the topology generated by

and

is the discrete topology on

. Consider the following statements:

I. In

, every infinite subset has a limit point.

II. The function

defined by

is a continuous function.

Which of the above statements is/are TRUE?

Your answerChoose one option.

Both I and II

I only

II only

Neither I nor II

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Let

and

be metric spaces, and let

be a continuous map. For any subset

, which one of the following statements is true?

Your answerChoose one option.

is open, then

is open

is connected, then

is connected

is closed, then

is closed

is bounded, then

is bounded

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GATE MA Topology PYQs

Question 41

Question 42

Question 1

Question 7

Question 37

Question 38

Question 11

Question 40

Question 42

Question 4