Assignment On The Topic Topological Spaces (unit-i)

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 x∈A 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 ? A  {x :

1  x  1} 2

,

D  {x :

1  x  1} 2

,

B  {x :

1  x  1} 2

E  {x : 0  x  1 and

,

C  {x :

1  x  1} 2

1  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 XY , A  B  A B . Q.7. Consider the lower limit topology on R and the topology given by the basis

M

={

[ 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 E = a

( 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.