GATE MA 2022 Paper 1 — page 8

Questions 22–24 of 65. Imported answers require review unless explicitly marked reviewed.

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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2022 · Paper 1

Question 22

Question 23

Question 24