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    BMSCW: I Semester M.Sc. Examination, January 2015 (Y2K11 (RNS) Scheme) Mathematics M103: Topology - I

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    rapsjade
    This document contains instructions and questions for a topology exam. It has two parts with multiple choice questions about fundamental concepts in topology, including: 1) Defining infinite and metric spaces, and proving properties like a superset of an infinite set being infinite and a subspace of a complete metric space being complete if closed. 2) Defining topologies and proving properties such as the intersection of two topologies being a topology, but the union not necessarily, and metrics inducing topologies. 3) Proving statements about open and closed sets in topological spaces, like a point belonging to a closure if every open set containing it intersects the set.

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    BMSCW: I Semester M.Sc. Examination, January 2015 (Y2K11 (RNS) Scheme) Mathematics M103: Topology - I

    Uploaded by

    rapsjade
    11 views2 pages
    This document contains instructions and questions for a topology exam. It has two parts with multiple choice questions about fundamental concepts in topology, including: 1) Defining infinite and metric spaces, and proving properties like a superset of an infinite set being infinite and a subspace of a complete metric space being complete if closed. 2) Defining topologies and proving properties such as the intersection of two topologies being a topology, but the union not necessarily, and metrics inducing topologies. 3) Proving statements about open and closed sets in topological spaces, like a point belonging to a closure if every open set containing it intersects the set.

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    Topology 15

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    This document contains instructions and questions for a topology exam. It has two parts with multiple choice questions about fundamental concepts in topology, including: 1) Defining infinite and metric spaces, and proving properties like a superset of an infinite set being infinite and a subspace of a complete metric space being complete if closed. 2) Defining topologies and proving properties such as the intersection of two topologies being a topology, but the union not necessarily, and metrics inducing topologies. 3) Proving statements about open and closed sets in topological spaces, like a point belonging to a closure if every open set containing it intersects the set.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    11 views2 pages

    BMSCW: I Semester M.Sc. Examination, January 2015 (Y2K11 (RNS) Scheme) Mathematics M103: Topology - I

    Uploaded by

    rapsjade
    This document contains instructions and questions for a topology exam. It has two parts with multiple choice questions about fundamental concepts in topology, including: 1) Defining infinite and metric spaces, and proving properties like a superset of an infinite set being infinite and a subspace of a complete metric space being complete if closed. 2) Defining topologies and proving properties such as the intersection of two topologies being a topology, but the union not necessarily, and metrics inducing topologies. 3) Proving statements about open and closed sets in topological spaces, like a point belonging to a closure if every open set containing it intersects the set.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 2

    *PG967* PG – 967

    I Semester M.Sc. Examination, January 2015


    (Y2K11 (RNS) Scheme)
    MATHEMATICS
    M103 : Topology – I

    Time : 3 Hours Max. Marks : 80

    Instructions : 1) Answer any five full questions choosing atleast two from
    each Part.
    2) All questions carry equal marks.

    PART – A

    1. a) Define an infinite set. Prove that every super set of an infinite set is infinite. 8
    W
    b) Let X be an infinite set, and x 0 ∈ X then prove that X – {x0} is an infinite set. 8
    SC

    2. a) Prove that the open interval (0, 1) of reals is non-denumerable set. 8

    b) State and prove Cantor’s theorem. 8


    BM

    3. a) Define a metric on a nonempty set X. If d is a metric on X, then prove that


    d( x , y )
    e(x, y) = , ∀x, y ∈ X is a metric on X. 8
    1 + d( x , y )

    b) Prove that a subspace of a complete metric is complete iff it is closed. 8

    4. a) State and prove contraction mapping theorem. 6

    b) State and prove Baire’s category theorem. 10

    PART – B

    5. a) Define a topology on a non-empty set. Prove that the intersection of two topologies
    is again a topology. 5

    b) Is the union of two topologies a topology ? Justify. 3

    c) Prove that every metric space is a topological space. 8

    P.T.O.
    PG – 967 -2- *PG967*

    6. a) Prove the following hold in [x, ℑ ]. 6


    i) α(φ ) = φ
    ii) A ⊆ B implies d(A) ⊆ d(B)
    ii) d(A ∪ B) = d(A) ∪ d(B)

    b) If A ⊆ (X, ℑ ], then prove that A ∪ d(A) is closed. 4

    c) Prove that a point x belongs to the closure of a set A iff every open set G
    which contains x has a nonempty intersection with A. 6

    7. a) Prove the following :


    i) A° ⊆ A
    ii) A is open iff A = A°
    iii) A ⊆ B ⇒ A ° ⊆ B°
    iv) A ° ∩ B° = (A ∩ B )°
    W 8
    SC
    b) Let (Y, ℑ *) ⊆ (X, ℑ ) and E ⊆ Y ⊆ X, then prove the following

    i) Ey = E ∩ Y
    BM

    ii) E° = E° Y ∩ Y °
    iii) bY(E) ⊆ b(E) ∩ Y 8

    8. a) Prove that a mapping f : X → Y is continuous iff inverses of open sets are


    open. 8

    b) Prove that a bijective function f : X → Y is a homeomorphism iff


    f( A°) = f(A)° ∀ A ⊆ X . 8

    ——————

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