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1) Show that a space X is T0 iff the closures of every two distinct
points are not equal.
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2) Show that the property of being a To- space is topological.
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3) Show that a finite cofinite topology is T1.
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4) Prove that a space X is T1 iff every singleton of X is closed.
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5) Show that every finite T1 - space is discrete.
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6) Show that the property of being a T1 - space is hereditary.
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7) Give an example to show that the image of a T1 - space under a
continuous mapping need not be T1.
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8) Prove that the property of being a T1-space is topological.
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9) Prove that a finite subset of a T1 - space has no limit point.
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10) Give the implication between T0, T1, T2, T3 and T4. And by
examples show that the converse is not true.
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11) Prove that a space X is T2 iff for each two distinct points, there
exists an open set U ∁ X containing one of them, and the other is
contained in the complement of the closure of U.
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12) Show that the property of being a space is Hausdorff is
hereditary.
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13) Let τ be the topology on the real line R generated by the open –
closed intervals (a , b]. Show that (R, τ) is Hausdorff .
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14) Show that every T3 - space is Hausdorff .
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15) In a regular space X, prove that the closures of any two
singletons either equal or disjoint.
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16) Show that the property of being a regular space is hereditary.
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17) Show that the property of being a regular space is topological.
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18) Prove that a space X is normal iff for each two disjoint closed
subsets F1, F2 of X, there exists an open set U ∁ X such that F1 ∁
U ∁ Ū ∁ X – F2.
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19) Prove that the property
of
being a normal space is weak
hereditary. Give an example to show that it is not hereditary.
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20) Prove that every T4 - space is completely Hausdorff .
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21) Show that the inverse image of a completely Hausdorff space
under an injective continuous mapping is completely Hausdorff.
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22) Prove that if f is a precontinuous injection from a space X into
a completely Hausdorff space Y for each open set V ∁ Y,
( f-1(V))-о ∁ f-1(V), then X is Hausdorff.
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23) Give the implications between T i΄ and Ti spaces, i ϵ {0, 1, 2}.
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24) Prove that a To΄ - space which T1 is a T 1΄- space.
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25) Prove that a regular To΄ - space is T 1΄.
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26) Show that the compact subsets of a discrete space are the finite
subsets.
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27) Show that the cofinite topological space is compact.
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28) Prove that the property of being a compact space is absolute.
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29) Prove that the every closed subset of a compact space is closed.
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30) Show that compactness is a topological property.
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31) Prove that a space X is compact iff
every class of closed
subsets of X which satisfies the finite intersection property has
itself a non-empty intersection.
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32) Prove that the property of being a Lindelöf
space
is
topological.
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33) Show that every closed subset of a Lindelöf space is Lindelöf.
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34) Prove that every compact Hausdorff space is normal.
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35) Prove that if f is a bijective continuous mapping from a
compact space X onto a Hausdorff space Y , then f is
homeomorphism.
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36) If X = {a, d, c, d, e} with topology τ = {X, ϕ, {a, b, c}, {c, d, e},
{e}}. Show that A = {a, d, e} ∁ X is disconnected.
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37) If A and B are non-empty separated subsets of a space X. Show
that A ∪ B is disconnected.
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38) Show that: A subset A ∁ X is connected iff it is not the union of
two non-empty separated sets.
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39) If A and B are connected sets which are not separated. Show
that A ∪ B is connected.
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40) Let A be a connected subset of X and let A ∁ B ∁ 𝐴. Show that
B is connected and hence 𝐴 is connected.
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41) Show that a space X is connected iff X, ϕ are the only subsets
which are both open and closed.
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42) Prove that connectedness is an absolute property.
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43) If f is a continuous mapping from a connected space X into a
space Y. Show that f (X) is connected.
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44) If A is a proper subset of a connected space X. Prove that:
b (A) ≠ ϕ.
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45) Prove that a confinite toplogical space X is connected.
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46) Let τ = {X, ϕ, {a}, {c, d}, {a, c, d}, {b, c, d, e}} be a topology
on X = {a, b, c, d, e}. Find the components of X.
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47) Show that every component of a space X is closed.
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48) For a subset A of a space X, show that
( X  A)  X  A .
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49) Prove that a mapping f from a space X into a space Y is
continuous if and only if the inverse image of each closed set in
Y is closed in X.
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50) Let Y be a subset of a topological space
 *  {Y  U : U  }
( X , ) ;
Show that
is a topology on Y.
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51) Let
X  {a, b, c, d , e}
and let A  {{a, b, c},{c, d },{d , e}} . Find the
topology on X generated by A .
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52) Let X be an infinite set. Show that   { , A : ( X  A) is finite} is a
topology on X.
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53) Prove that a mapping f from a space X into a space Y is
continuous if and only if
f ( A )  f ( A)
for every A  X .
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54) Let A be a subset of a space X, show that
{ A , Ab , ex ( A)}
is a
partition of X .
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55) Let En = {n, n + 1, n + 2, …}, n N. Prove that  = {  , En: n
N} is a topology on N. List the open sets containing 4. If A = {7,

24, 47}  N, find A, A , A , A .
b
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56) If  and  are two topologies on X, then prove that    is a
topology on X.
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57) Consider X = {a, b, c, d, e} with the topology  = { X,  , {a},
{a,b}, {a,c,d}, {a,b,e}, {a,b,c,d}} and A = {a,b,c}. Find
A, A , A , Ab , ex( A)
and the relative topology  A on A.
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58) Let X be a space. Then show that X is a To- space iff
{x}  {y}
for each x, yX, x  y.
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59) Let X be a space and p

X show that the intersection N  W of
two neighborhoods n, W of p is also a neighborhood of p.
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60) Give an example for an open mapping which is not continous.
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61) Let Y be a subset of a topological space
 *  {Y  U : U  }
( X , ) ;
Show that
is a topology on Y.
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62) Let (X, ) be a topological space and A  X. Show that: A is
closed iff A contains all of its limit points.
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63) Let f be a mapping from a space X into a space Y, show that: f
is open iff
f ( A )   f ( A) for
each A  X .
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64) Let f be a mapping from a space X into a space Y, show that: f
is closed iff
f ( A)  f ( A )
for each A  X .
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65) Let f be a bijective mapping from a space X onto a space Y,
show that: f is a homeomorphism iff
f ( A)  f ( A ) for
each A  X .
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66) Let (X, ) be a topological space and A  X. Show that
A  A  Ab .
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67) Show that the union of two preopen subsets of X is preopen.
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68) Define:
T0 , T1 , T2
and give the implications between them.
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69) Prove that any superset of a neighborhood of
p X
is also a
neighborhood of p .
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70) Let X be a space and
A, B  X ;

show that  A  B   A  B .
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71) Let

be
the
set
of
real
numbers
and
U  { A   : p  A, an open interval S st : p  S  A}. Prove that (, U ) is
a topological space.
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72) Show by example that any arbitrary intersection of open sets in

need not be open.
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73) Give an example for a continuous mapping which is not open.
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74) Prove that a subset A of a space X is closed iff A contains all of
its limit points.
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75)Give the implication between T0 , T1 and T2 ; and by examples
show that the converse is not true.