Topology-I - BU 2023 Question Paper

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MN-443

M.A./M.Sc. I^st Semester (Reg./Pvt./ATKT)

Examination, 2023-24

Maths

Paper - III

Topology-I

Time : 3 Hours

[Maximum Marks : Reg.=85 Pvt.=100]

Note:- Attempt all the questions.

SECTION - 'A'

Choose the correct answer :

(i) The set of real numbers is:

(a) Countable (b) Uncountable (c) Countably Infinite (d) None of these

(ii) Give the statement of Cantor's Theorem :

(iii) The empty set φ and the full space X are :

(a) Both open (b) Both closed (c) Both open and closed (d) None of the above

(iv) The boundary of $A \subseteq X$ consists of all points X in X with the property that :

(a) Each Neighbourhood of X intersects A but not Aⁱ (b) Each Neighbourhood of X intersects both A and A^c (c) Each Neighbourhood of X intersects A' but not A (d) None of these

(v) Write down the Kuratowski closure axioms :

(vi) Which of the following is not true :

(a) Every open interval is an open set (b) Every open interval is a neighbourhood of each of its points (c) Every closed interval is a neighbourhood of each of its points (d) None of these

(vii) Define first and second countable space :

(viii) Define a separable space.

(ix) The set Q all rational numbers are :

(a) Connected (b) Not Connected (c) Path Connected (d) None of these

(x) Each interval and each ray in the real line is :

(a) Connected (b) Locally Connected (c) Connected and both Locally Connected (d) None of the above

SECTION - 'B'

Short Answer Type Questions 5×5=25

Define:

(a) Countable set (b) Well-ordered set

Give the statements of:

(a) Cantor's Theorem (b) Zorn Lemma

Define topological space and give example of it.

Let X be a topological space and A an arbitrary subset of X. Prove that $\bar{A}=\{x;$ each neighbourhood of x intersects A$\}$.

Let X be a topological space and A a subset of X. Then prove that :

(a) $\bar{A}=A \cup D(A)$ (b) A is closed $\iff A \supseteq D(A)$

Let X be a space and for $x \in X, N_x$ be the neighbourhood system at x then prove that :

(a) If $U \in N_x$ then $x \in U$ (b) For any $U, V \in N_x \cup V \in N_x$ (c) If $V \in N_x$ and $U \supset V$ then $U \in N_x$

Let X be a second countable space. If a non-empty open set G in X is represented as the union of a class $\{G_i\}$ of open sets then prove that G can be represented as the countable union of $G_i$'s.

Let X be a second countable space. Then prove that any open base for X has a countable subclass which is also an open base.

Prove that the image of a connected space under a continuous map is connected.

Define locally connected and locally path connected space.

SECTION - 'C'

Long Answer Type Questions 9×5=45

Prove that a finite product of countable sets is countable.

If X and Y are two sets each of which is numerically equivalent to a subset of the other, then prove that all of X is numerically equivalent to all of Y.

Define the following :

(a) Relative Topology (b) Continuous Map (c) Open Map (d) Homeo Morphism

Prove that the real line R and complex plane are separable.

Let $T_1$ and $T_2$ be two topologies on a non-empty set X, then show that $T_1 \cap T_2$ is also a topology on X.

Let X be a non-empty set, and let there be given a "closure" operation which assigns to each subset A of X a subset $\bar{A}$ of X in such a manner that (1) $\phi = \bar{\phi}$ (2) $A \subseteq \bar{A}$ (3) $\overline{\bar{A}} = \bar{A}$ and (4) $\overline{A \cup B} = \bar{A} \cup \bar{B}$. If a "closed" set is defined to be one for which $A = \bar{A}$, then prove that the class of all complements of such sets is a topology on X whose closure operation is precisely that initially given.

10.

Prove that every separable metric space is second countable.

Prove that a subset of a topological space is open if and only if it is a neighbourhood of each of its points.

11.

Prove that a space X is locally connected if and only if for every open set U of X, each component of U is open in X.

Prove that the union of a collection of connected subspaces of X that have a point in common is connected.