Kannur University 2011-1st Sem M.Sc Mathematics ( Syllabus) MODEL MAT1C04: Topology- - Question Paper

PART-A
1. Let ? be the set of positive integers, ?? be the discrete metric on ?
Let ??: ? ?? ?? be described by ?? ????? = | ?
? ??
? | for ??? ?. Prove that ?? is a
metric on ?. Show that ?? and ?? induces the identical topology on ?.
2. Prove that in the co-countable topology the only convergent sequences are those which
are eventually constant.
3. Let A = ???????
????????? ? ?? and let ? = A where the closure is w.r.t the usual
topology on ??G Prove that ? is connected.
4. describe quotient map; show that every closed surjective continuous function is a quotient
map.
5. Let ? be a Hausdorff space. Prove that for any space ? and any 2 continuous
functions ??? ?? ??, the set ? ? ?? ?????? ????? is closed in ?.
PART-B
6. describe 2nd countable space. If ? is a 2nd countable space, show that every open
cover of ? has a countable subcover.
7. a) Show that metrizability is a hereditary property.
b) Show that continuous image of a compact space is compact.
8. Prove that every continuous real - valued function on a compact space is bounded and
attains its extrema.
9. Characterize connected subsets of ? in the usual topology.
10. For a topological space ?????, prove that the subsequent statements are equivalent.
i. ? is a ?? – space.
ii. For any ? ??? the singleton set ??? is closed.
iii. Every finite subset of ? is closed.
iv. The topology ? is stronger than the cofinite topology on ?.
11. Show that a continuous bijection from a compact space onto a Hausdroff space is a
homeomorphism.
PART- C
UNIT I
12. a) describe subspace topology. Let ? be the topology on ?? induced by the euclidean
metric. Show that the subspace topology on ? induced by ? is identical as the usual
topology on ?
b) Let ? be a set and ? be a family of its subsets covering ?. Prove that the subsequent
statements are equivalent.
i. There exists a topology on ? with ?as a base.
ii. For any ????? ????? ??? can be expressed as the union of a few numbers of ?.
iii. For any ????? ?? and ? ??? ??? there exists ?? ? ? such that ? ? ?? and
?? ? ?? ???G
13. a) For a subset of A of a topological space ?, let A denote the closure of ? and ??denote
its derived set. Show that ???? ???
b) Let ??????????? be topological spaces and ????? a function.show that the
subsequent statements are equivalent.
i. ? is continuous on ?
ii. For all ?? ?? ?????? ? ?
iii. For all ? ???????? ???????????
UNIT-II
14. a) Let ? be a collection of connected subsets of a topological space ? such that no
two elements are mutually separated. Show that ?? ??? connected.
b) describe Locally connected Space. provide an example of a space which is connected but
not locally connected.
15. a) Show that every quotient space of locally connected space is locally connected.
b) Show that every path-connected space is connected. Is the converse true? Justify your
answer.
UNIT-III
16. a) Show that for a topological space????? the subsequent are equivalent
i) The space ? is regular
ii) For any ? ?? and any open set ? containing ? there exists an open set ?
containing ? such that ?? ??.
iii) The family of all closed neighbourhoods of any point of ? forms a local base at
that point.
b) Show that every metric space is ??.
c) Prove that regularity is a hereditary property.
17. a) Let ????? be a topological space , ????? a metric space and ???? a sequence of
continuous functions from ? ?? ? which converges uniformly to ? .Show that ? is
continuous.
b) Show that every continuous real-valued function on a closed subset of a normal space
can be extended to the whole space.

Earning:   Approval pending.