Topology-I

(i) Which is not true in the following :

(a) The set of all even positive integers is countable

(b) The set of all even integers is countable

(c) The set {1¹, 2², 3³, .....} is countable

(d) None of these

(ii) Which is true in the following :

(a) The set of integers is well ordered in the usual order

(b) The set of integers is not well-ordered in the usual order

(c) The set {x \(\in\) R / 0 \(\leq\) x \(\leq\) 1} is well-ordered in the usual order

(d) None of these

(iii) The total number of topologies defined on the set X = {a} is

(a) 1

(b) 2

(c) 3

(d) None of these

(iv) The boundary of A \(\subset\) X consists of all points x in x with the property that.

(a) Each neighborhood of x inter section A but not A^l

(b) Each neighborhood of x intersects A^l but not A

(c) Each neighborhood of x intersects both A and A^l

(d) None of these

(v) The boundary of A is

(a) A \(\cap\) \(\overline{A}\)

(b) \(\overline{A}\) \(\cap\) (\(\overline{A}\))^l

(c) \(\overline{A}\) \(\cup\) (\(\overline{A}\))^l

(d) None of these

(vi) Which is not true in the following.

(a) Every open interval is an open set

(b) Every open interval is a neighborhood of each of its points

(c) Every closed interval is a neighborhood of each of its points

(d) None of these

(vii) Which is true in the following :

(a) A limit point of a set A is also an isolated point of A

(b) An isolated point of a set A belongs to A itself

(c) An isolated point of a set A is also a limt point of A

(d) None of these

(viii) Which is not ture in the following :

(a) \(\overline{A}\) is the intersection of all closed subsets of A

(b) \(\overline{A}\) is the intersection of all closed supersets of A

(c) \(\overline{A}\) is the intersection of all supersets of A

(d) None of these

(ix) Which is not ture in the following :

(a) Every path connected space X is connected

(b) The continuous image of path connected

(c) A space is path connected it is connected

(d) None of these

(x) Which is true in the following :

(a) Each interval in the real line is both connected and locally connected

(b) The rationals Q are connected

(c) The rationals Q are locally connected

(d) None of these

Explain

(a) Axiom of choice

(b) Well ordered set

Define

(a) Continuum hypothesis

(b) Counably in finite set

Define sub space of a topological space and give one example of it.

Show that a subset of a topological space is dense \(\Leftrightarrow\) it intersects every non empty open set.

Define kuratowski's closure axiom.

Show that int (\(\overline{A}\)) = (A) for every subset A of a topological space X.

Define open base for a topological space. Give an example of it.

Prove that every second countable space is separable.

Define connected space. What do you mean by a disconnection of a topological space X.

Prove that a topological space X is disconnected there exists a continuous mapping of X on to the discrete two-point space {0, 1}.

Prove that the set Q⁺ of positive rational numbers is countable infinite.

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.

If T₁ and T₂ are two topologies on a non empty set X, then prove that T₁ \(\cap\) T₂ is also a topology on X.

Let x,y and z be topological space. If f : x \(\to\) y and g : y \(\to\) z are continuous mappings, show that if g \(\circ\) f : x \(\to\) z is also continuous.

Let X be a topological space and A an arbitrary subset of X. Then prove that \(\overline{A}\) = {x : each neighborhood of x intersects A}.

Let f: x \(\to\) y be a mapping of one topological space into another. Show that f^-1(I) is closed in X whenever I is closed in Y.

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

Let f : x \(\to\) y be a mapping of one topological space into another and let there be given on open base in y. Then prove that f is continuous \(\Leftrightarrow\) the inverse image of each basic open set is open.

Let X be a topological space and A a connected subspace of X. If B is a subspace of X such that A \(\subseteq\) B \(\subseteq\) \(\overline{A}\) then prove that B is connected.

Let X be a locally connected space. If Y is an open subspace of X, then prove that each component of Y is open in X.