Math Chapter Wise

EnglishOPSC PGT 2017

Q21.Helne-Borel property ensures that:

AAn open cover of a compact set has a finite sub-cover BAn open subset of a compact set has a bounded sub-cover CUniformly continuous function is differentiable. DUniformly continuous is Riemann integrable

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EnglishOPSC PGT 2017

Q22.In a complete metric space:

AEvery infinite series is convergent BEvery subset is compact CEvery function is closed DEvery Cauchy sequence is convergent

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EnglishOPSC PGT 2017

Q23.For metric space X with metric d, the map is: X X is a contraction of X, if, for x, y ∈ X:

Ad((x), (y)) cd(x, y) with finite positive c Bd((x), (y)) cd(x, y) with real c ≤ 1 Cd((x), (y)) cd(x, y) with 0 < c < 1 Dd((x), (y)) cd(x, y) with real c < 1

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EnglishUPPSC GDC 2021

Q24.In a topological space (X, T), Where X = {a, b, c} and T = {φ, {a},{b, c}, X}. Which of the following is true?

ASets in T are open but not closed BSets in T are closed but not open CSets in T are both open and closed DSets in T are neither open nor closed

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EnglishUPPSC GDC 2021

Q25.Every regular Lindelof space is

AI countable BII countable CNormal DNone of the above

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EnglishUPPSC GDC 2021

Q26.A subspace of a normal space is

ANormal BHausdorff CNeed not normal DClosed

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EnglishUPPSC GDC 2021

Q27.Let X = {a, b, c} and T = {φ, {a},{a, b}, X} be a topology on X. Then (X, T) is

ACompact only BConnected only CBoth compact and connected DNeither compact nor connected

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EnglishUPPSC GDC 2021

Q28.Which one of the following is NOT a topology on X = {a, b, c}?

A{,{a},{b,c},X} φ B{,{b},{a,c},X} φ C{,{a},{b},{a,b},X} φ D{,{a},{b},{c},{a,b},X} φ

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EnglishUPPSC GDC 2021

Q29.Let X = {a, b, c} and T = {φ, {a}, {a, b},{a, c}, X} be a topology on X. If A = {a, c}, then derived set of A is

Aφ B{b, c} C{a, c} D{a, b}

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EnglishUP Higher Asst. Prof. 2021

Q30.A subset Y of a topological space (X, τ τ τ) is no where dense if -

AY has empty interior BY has empty interior CClosure of interior Y is non empty DNone of these

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EnglishUP Higher Asst. Prof. 2021

Q31.Any infinite subset A of a discrete topological space X is,

Anot compact Bcompact Ccompact and connected Dconnected only

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EnglishUP Higher Asst. Prof. 2021

Q32.Metric d is known as pseudo metric if -

Ad(x, y) = 0 ⇒ x = y Bd(x, y) = 0 ⇔ x = y Cx = y ⇒ d(x, y) = 0 Dnone of these

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EnglishUPPSC GDC 2021

Q33.Which of the following is a topology on X = {a, b, c, d}?

A{ },{a,{b},{b,c},X} φ B{ },{a},X φ C{ },{a,{b},X} φ D{ },{c,{d},{b,c},X} φ

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EnglishUPPSC Polytechnic Lecturer 2021 (I) UP Higher Asst. Prof. 2021

Q34.Which of the following in not correct?

A1 0 T – space T – space ⇒ B2 0 T – space T – space ⇒ C2 3 T – space T – space ⇒ D4 2 T – space T – space ⇒

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EnglishUPPSC Polytechnic Lecturer 2021 (I)

Q35.A subset 'A' of a topological space 'X' is said to be not dense if-

Aand BImage/formula option B (see source) CA = φ DBoth

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EnglishUPPSC Polytechnic Lecturer 2021 (I)

Q36.Consider { } { } { } { } φ 1 J =,X, a, b, a,b and { } { } { } φ 2 J =,X, a,b, b,c. Which of the above is/are a topology on { } X = a,b,c?

AOnly J1 BOnly J2 CBoth J1 and J2 DNeither J1 noe J2

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EnglishUPPSC Polytechnic Lecturer 2021 (I)

Q37.The number of non-trivial topologies on a set { } X = a,b,c is–

A27 B28 C29 D30

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EnglishUPPSC Polytechnic Lecturer 2021 (I)

Q38.Let X = {a,b,c} and let { } φ J =,{a},{b,c},X, then

A(X,J) is normal B(X,J) is T1 C(X,J) is T2 D(X,J) is T3

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EnglishUPPSC Polytechnic Lecturer 2021 (I)

Q39.In which of the following cases, there is no continuous function 'f' from the set 'S' onto set 'T'?

AS=[a,b],T=R BS=(a,b),T=R CS=(a,b),T=(a,b] DS=R,T=(a,b]

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EnglishUPPSC Polytechnic Lecturer 2021 (I)

Q40.Let (X,J) be a topological space, where X={1,2,3,4,5} and J= { } φ,X,{2},{3,4},{2,3,4},{1,2,3,4}. If A= {1,3,4}, then interior of A is-

A{1,4} B{3,4} C{2,3} D{2,4}

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