8/14/2019 Assignment on the Topic TOPOLOGICAL SPACES (UNIT-I)

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Assignment on the Topic: Topological spaces (Unit-I )

Q.1. Let X be a topological space, let A be a subset of X .Suppose that for each xA

there is an open set U containing x such that U A. Show that A is open set in X.

Q.2. Show that the collection A = { ( a , b ) : a < b , a , b Q} is a basis that

generates the standard topology on R.

Q.3. Show that the collection M = { [ a , b ) : a < b , a , b Q } is a basis that

generates a topology different from the lower limit topology on R.

Q.4. Consider the set Y = [ -1, 1] as a subspace on R . Which of the following sets

are open set in Y ? Which are open set in R ?

1{ : 1}2

A x x , 1{ : 1}2

B x x , 1{ : 1}2

C x x

1{ : 1}

2 D x x ,

1{ : 0 1 } E x x and Z

x

Q.5. Show that if U is open set in X and A is closed set in X , then U A is open set in

X and A U is close set in X .

Q.6. If A X and B Y . Show that in the space XY , A B A B

.

Q.7. Consider the lower limit topology on R and the topology given by the basisM

= {

[ a , b ) : a < b , a and b are rational} . Determine the closure of the interval A = (0, 2)

and B = ( 2,3) in these topologies .

Q.8. Let be the topology on R consisting of R , and all open infinite intervals Ea =

( a , ) , where a R . Find the interior, exterior and boundary of the set A = [ 7 , ) .

Q.9. Let be the topology on N consisting of and subsets of N of the form En =

{ n , n + 1 , n + 2 , .. } , where n N .(i) Find the accumulation points of the set A = { 4 , 13 , 28 , 37 }

(ii) Determine those subsets E of N for which E = N .

Q.10. Show that lower limit topological space is First countable space but not second

countable space.

8/14/2019 Assignment on the Topic TOPOLOGICAL SPACES (UNIT-I)

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