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1 Every complete metric space is a
___________
Baire space
Blank space
Dense space
Cardinal space
2 Let
(X,τ)
be a topological space. If
X
is second countable, then
X
is
____________countable
Third
Second
Fourth
First
3 If
xϵA¯
, then there exists a sequence
(xn)
of A such that
xn→x
is only true if X is a(an)
___________________
Countable
Metrizable
Hausdorff
Separation
4 W is a neighbourhoodbasis of a
point
xϵX
if one of these holds except one.
∀wϵW,wϵN(x)
vϵN(x),thereexistswϵWsuchthatWϵV
A and B
∀xϵX,X=A¯
5 Let
X
be a topological space. Then the following hold for
X
except one
X
is countable if it contains a countable neighbourhood basis
X
is second countable if it contains a countable basis
X
is non-separableeven when it contains a countable dense subset
X
is separable if it contains a countable dense basis
6 Let
X
be a sequence
(xn)
of elements of a subset
of
A
of a topological space
X
, such that
xn→xϵX
, then
xϵX⋂Xc
xϵA
xϵA¯
xϵXn
7 Let
X=(a,b,c,d,e)andτ=(X,ϕ,[a],[c,d],[a,c,d],[b,c,d,e]).LetA=[a,c]
]
, then set
A’ of limit points of A
is given by
A′=(b,c,e)
A′=(b,d,e)
A′=(b,e)
A′=X
8 Let
R
, the real line be endowned with the discrete topology. Which of the following subsets
of
R
is dense in
R
Q
Ritself
Qc
All singletons
9 Let
A=(0,1]⋃2
be a subset of
R
. Then the isolated points
of
AinR
are
0 and1
0 and 2
1 and 2
[2]
10 For the aet \mathbb A in question (9) above, which of the following are the limit points of
A
?
0 and1
0 and 2
1 and 2
2 only