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General Topology
§1. Topological Space
1.1 Definitions of topology & topological space
Definition 1. Topology on set : T is a collection of subsets
in X, satisfied that a) Ø, X T ;
b) intersection of any
finite number of elements of T belongs to T ;
c) union of elements of any subfamily of T belongs to T .
Remark: T includes Ø, X, and closed under finite
intersection & arbitrary union .
Example 1 Anti-discrete topology : {Ø, X } (trivial topology).
Ex 2 Discrete topology :
2 X (Exp X) , all subsets of X.
Ex 3 Sierpinski topology: X={0,1}, T ={Ø, {0}, X }.
Definition 2. Topology space
are called open sets.
( X ,T );
the elements of T
For any set, normally we can define several different topologies
Definition 3 Let T , T 2 be the topologies on set X, if T 1  T 2 ,
T 1 is weaker or coarser than T 2 . T 2 is stronger or finer than T 1 .
1
Let T  :   A be a collection of topologies on X.
T = T  :   A=U  X :U T  for all   A is weaker than
all the T  .
1.2 Sub-bases & bases
LetE be a collection of subsets of a set X . LetBE be the
smallest collection of subsets of X which contains all the
elements of E together with the empty set and the whole
set X and which is closed under finite intersection. Let
T E { :  BE } , then
T E is a topology on X , the smallest topology containing E .
The collection E is called a subbase of the topology T E and
we say that the topology T E is generated by the subbase E .
Definition 3. A collection B of open subsets of a topology
space X is called a base of the space if each open set in X is a
union of some elements of B .
Since T is closed under arbitrary union . T itself is also
a base. We want to get a base of smaller cardinality.
Theorem B is a base of (X ,T ) if and only if all the
elements of B are open and for any x∈X, open set U containing
x, there exists V ∈B such that x ∈V 
U.
Proof :    For any x∈X , open set U containing x , U is the
union of some elements B , i.e. B ' B such that
U = V :V B ' , x∈U , then V B ', x ∈V. It is
clear that V U .
   For any open set U and any x U ,
xVx U , thus U  xU Vx. U.
Vx B
such that
Topology T is uniquely determined by its base B , and
T =  :  B 
As is known, single topology T can have different bases .
Uncountable T can have countable base.
Proposition 1. A collection B of subsets of a set X is a base of
a topology on X if and only if, B  X and for any U, V B
and any x U V , there exists W B such that x W  U V.
Proof : ( ) X  T , hence X  B . U ,V B  U  V . T
Therefore, for x U  V , W B , x W  U  V .
( ) Let T  U  X : BU B s.t. U = BU .；, X T
and T is closed under arbitrary union. For U , V T .
we prove that U V T . For any x U V , since U  BU
V = BV , U 1 ,V1 B such that x U 1 V1 , then Wx B
such that xWx U1 V1 . Thus U V = xU V Wx .
We let BU V  Wx , xU V  , then U V = BU V. This shows that
T is closed under finite intersection. T is a topology. It is easy
to see that B is a base for T .
Note: (1) A collection B of subsets of a set X is a base of
a topology on X if B  X , and B is closed under finite
intersection.
(2) BE in definition of sub-base is a base of T E .
Example 4. Given a set R. B
. ( a,b  , a  x isba base for
the usual topology T . One can show that U T if and only if
U is a union of pairwise disjoint intervals .
2
Example 5. Given a plane R 2 , B
for natural topology on plane,
where B  x   {(x,y): x 2 + y 2   } .
  B (x)：x R 
is a base
Example 6 Bs =a,b  : a  x  b . It is clear that Bs is closed
under finite intersection & Bs  R , form a base for T S .
The topological space ( R ,Bs ) is called the Sorgenfrey line.
1


(a, b) 
a , b , so

T S is stronger than the
n


usual topology T on the line R . T  T S and T  T S .
T S does not have a countable base.

n =1
B0 B
B0  P
B0
Proof : Let t= U,V  , U , V P . Let  :P P B , t   t 
  t   W such that V W U if such W exists; otherwise  (t)=  .
Let B0 (t) : (t)  . B0 B and B0 c P P c P . B0 is a
base for T .
How to prove that φ is surjective and B is a bsae?
We can prove it as follows: for any open set O and xO , there
exists U P such that xU  O . Choose W B with xW U
Choose V P with xV W U .We can do this since both P
and B are bases. By definition of  . Let t= U ,V  and
W =(t)B0 , then x  (t)  O .Thus B0 is a base since x , O are
arbitrarily chosen.
Corollary No countable subset  of Bs can be a base for  X,T s .
Let
A  {a : b R ,  a,b  } ,
if it is countable . Choose  c,d 
cR\ A and c  d , then c,d Bs can not be written as union
of elements in  .
If there is some a < c, then…. Otherwise, …
1.3 Neighborhood of points. Nearness of a point to a set and
the closure operator
Definition 4 A set O  X is called a neighborhood of a point x
in the topological space X if there exists an open subset U of X
such that xO U .
Remark 1. Open set is neighborhood of every point it contains.
Remark 2. Intersection of two nbds is again a neighborhood.
Neighborhood is a “measure” of nearness to the point , also it is an
approximation to the point. There is no distance in topological
spaces.
Definition 5. Let  X,T  be a topological space and let A  X
and x X .We say that the point x is near to(or at distance zero
from) the set A and write x A (at distance 0) if for every open
nbhd U of x , U A   .
Otherwise we say x is far from A ,write x A (or   x,A  1 ) .
 is said to be induced by topology T , to emphasize this we
write δ= δ(T ) .
Properties of  : x X ,
1) x (   x,   1 )
3) x  A B   x A  x B
A X
,
2)
B X
x A  x A
4) x B  yB y A  x A (weak form of triangle axiom)
In condition 3, if a point far from A B , then it is also far
from A and B . In condition 4 , the transitivity of the nearness
relation 0+0=0 .
The set of all points near to a subset A  X is called the closure
of A in the topological space X. Thus , A x X : x A .The map
A  A is called the closure operator in the topological space X.
The closure operator possesses the following properties
1)  
2) A  A for all A  X
3) A B  A B
4) A  A
Definition : The set which contains all points near to it is said
to be closed. A is closed if and only if A  A .
Proposition 3 A set A is closed in a topological space  X,T if
and only if X\A is open in  X,T 
Proof : A= A  {x : x A} For y  X \ A , there exists a nbhd U of
y,such that U A= , hence y U  X \ A
 every point y
in X\A is its interior point  X\A is open.
By De Morgan law, closedness is closed under finite union &
arbitrary intersection.
Note that: A is a closed set ( A  A). and is the smallest
closed set containing A.
Example 7. Let  X,T  be an anti-discrete space. T =,X  .
For any nonempty A, A  X . In discrete space, T  2 X , A= A .
In Sorgenfrey line  R ,T s  , R  R .
Proposition 4. If U and V are disjoint open subsets of X , then
U V   ( and V U   )
Proof: For any xV , V itself is an open neighborhood of x
and V U =  . Hence xU by the definition, U V  .
1.4 Definition of a Topology using a Nearness relation or a
closure operator
An (abstract) closure operator on X is a rule which assigns to
each subset A of X . Map 2 X  2 X, A A , satisfies
1)  
2) A  A for all A  X
3) A B  A B
4) A  A
Closure operator can be obtained via (abstract) nearness relation
by defining A x X : x.A
Let F   A  X  A= A ,then T  X \ A AF  is a topology on X .
Proposition 5. The topology of a space X is uniquely determined
by any one of the following objects: the nearness relation  , the
closure operator , or the collection of all closed sets.
Proof: Only for closure operator. Let T  X \ A AF 
1)  X \ = X T ; X  X hence X  X , X\ X T .
2) A,B T  X \ A  X \ A  X \ B  X \ B
X \  A B    X \ A  X \ B   X \ A X \ B   X \ A   X \ B 
= X \ A B . Hence A BT .
3) Suppose that G  T . For any AG , X \ A  X \ A . On the other
hand for any A1 G , we have X \  AG A  AG  X \ A  X\ A1 .
Hence X \  AG A  X\ A1  X\ A1 and
X \
AG
A 
It is clear that
X \
AG
A  X \ 
A1G
X \
AG
 X \ A1   X \ 
AG
A
A  X \ 
,
AG
A1G
AG
A
AT
A1   X \ 
AG
A
, therefore
.
Thus T is a topology on X , and - is the closure operator. uniquely determines the closed subsets in X, hence uniquely
determines its topology .
Definition. For any set A  X , the derived set of A is denoted
by d(A) , it consists of all its limit points (accumulation point).
A point x is called to be a limit point of A if for any neighborhood
O of x , O  A\x .
We have that A  A d  A by definition of the derived set.
Four properties of derived set
1) d    
3) d  A B   d  A d  B 
2) A  B  d  A  d  B 
4) d  d  A   A d  A  A
The above four axioms uniquely determined a topology T such
that in this topological space, the derived set of A is exactly d(A) .
The relation of interior , closure and complement
, -,'
IntA  A  x  A|  an open nbhd O of x , O  A  , which is an open set
Remark: (1) A is open iff A = Int(A); (2) Int(A) is the union of
all open sets contained in A.
1) 
3)  A
 , X X
B  A B
2) A  A
4)  A   A
Theorem 1. A  A' - ' , A-  A' ' .
Proof : For any x  A , A is a neighborhood of x. Since A A'  ,
then x  A' -  x  A' - ' . Thus A  A' - ' .
On the other hand , for any x  A' - ' , we have x  A' - . Then
there exists a neighborhood V of x , such that V A'  V  A .
That is A' - '  A .
Theorem 2 (C.Y.Yang) d(A) is closed
d{x} is closed. P 77 6.
Û
Theorem 3 (Kuratowski)
P 81 4.
Definition 2 Boundary set of A, Bd(A):
Bd (A)= ¶ (A)= A
\A .
Bd ( A) = A \ A = A Ç ( X \ A ) = A Ç A'- = Bd ( A¢).
Bd ( A) = A Ç ( X \ A)- .
Definition 3. The neighborhood system U x of point x : the
collection of all x's neighborhoods. It satisfies
1) For any U x  , U U . xU
2) Closed under finite intersection: U,V U x , U V U .
3) Closing up : U U x , U V V U .
4) Contains an open neighborhood V (V is a neighborhood of all
its points) U Ux  V Ux  V  U  y V  V U y  
Neighborhood system uniquely determine a topology T and in
(X,T ) , the neighborhood system coincide with pre-assigned one.
1.5
Subspaces of topological space
To each subset Y of a topological space (X, T ) associated a
new topological space Y , T |Y  where T |  U Y : U  T  is the
set of all “traces” in Y of the open subsets of X. T |Y is said to
be the topology generated(or induced) by T and Y , T |Y  is
called a subspace of (X, T ) .
Y
Properties of subspace can be very different from the whole
space.
Any figure on the plane: disk, circle, disk with a hole.
Q, J rational & irrational number with induced usual topology.
Thm. If B is a base of (X, T ), then B ' =
base of the subspace (Y, T Y ).
{B Ç Y : B Î B }
is a
Proof: (1) Obviously, each U  Y  B  is open in Y.
(2) Suppose x  U and U is open in Y, we have U = V  Y
where V is open in X. By the definition of bases,  B  B such
that x  B  V. It follows that x  B  Y  V  Y = U.
Thm. Let Y be a subspace of X. If U is open in Y and Y is
open in X, then U is open in X.
Relative nature of closed set . A topological space (X, T ) ,
(Y , T Y ) is a subspace. (1) A Ì Y can be closed in Y, but not in X.
(2) clY (A)= clX (A)Ç Y for A Ì Y Ì X .
A = cl X (A)
Proof: (1) Let x  cly(A), we prove that x  clx(A). For each
nbhd U of x in X, U  Y is a nbhd of x in Y. So, (U  Y)  A =
. That is U  A  . Therefore, x  clx(A)  Y.
Let x  clx(A)  Y. For each nbhd V of x in Y,  nbhd U of x in
X s.t. V = U  Y. From x  clx(A), U  A  . This implies
that (U  Y)  A  , i.e., V  A  . Thus, we conclude that
x  cly(A).
Thm. Closed family in Y := {F Ç Y : F is closed in X }, using
subspace we can construct many interesting examples.
Proof: (1) Let A be closed in Y, then Y \ A is open in Y. So, 
open set U in X s.t. Y \ A = U  Y. It is easy to check that A = Y
 (X \ U), and X \U is closed in X.
(2) Assume that F is closed in X, then X \ F is open in X. So,
(X \ F)  Y is open in Y. Since (X \ F)  Y = Y \ (F  Y), we
know that F  Y is closed in Y.
Example 8. (The Cantor perfect set) We use the word segment
to refer to a set of the form [ a, b] and interval to refer to a set of
the form (a, b) . A1= segment [0,1] , An is a union of a finite
number of pairwise disjoint segment. An + 1denote set obtained by
removing from each segment of An the middle 13 interval.
Then we will get a decreasing sequence An : n ΠN . C = ǥn= 1 An
of the line R is called the Cantor perfect set . C is closed, has no
isolated points, but does not contain any interval , uncountable and
measure 0.
Remark: A point of the set A \ d(A) is called an isolated point
of A. A point x is an isolated point of X iff the one-point set {x}
is open.
1.6 The Product Topology
Def. Let ( X , T X ) and (Y , T Y ) be topological spaces. The product
topology on X ´ Y is the topology T having as base the collection
{U ´ V : U Î T X ,V Î T Y }
Prop: If B is a base of the space ( X , T X ) andC is a base of the
space (Y , T Y ) , then the collection {B ´ C : B Î B , C Î C } is a base of
the product space (X, T ) .
Def: Let p1 : X ´ Y ® X be defined by the equation p1 ( x, y) = x
Let p 2 : X ´ Y ® Y be defined by the equation p 2 ( x, y) = y
The map p1 and p 2 are called the projections of X ´ Y onto its
first and second factors, respectively.
Thm. The collection S  { 11 (U ) : U  T X }  { 2 1 (V ) : V  T Y }
is a subbase of the product topology on X  Y.
Proof: Let T be the product topology and T  the topology
generated by S.
(1) Because S  T , we have T   T.
(2) On the other hand, every base element U  V of the
topology T has the form U V  11 (U )   2 1 (.V ). Thus, U  V
 T . So that T  T  as well.
1.7 The metric topology
Def: A metric on a set is a function d: X  Y  R having the
following properties:
(1)d(x, y)  0 for all x, y  X; equality holds iff x = y.
(2)d(x, y) = d(y, x) for all x, y  X.
(3)d(x, y) + d(y, z)  d(x, z), for all x, y, z  X. (triangle inequality)
Def: Given a metric d on X, the number d(x, y) is called the
distance between x and y. Given  > 0, the set {y: d(x, y) < }
is called the –ball centered at x. It is usually denoted by B(x, ).
Def: If d is a metric on the set X, then the collection of all balls B(x, ), for x  X and  > 0, is a base of a topology on X,
called the metric topology induced by d.
Def: If X is a topological space, X is called to be metrizable if there
exists a metric d on the set X that induces the topology of X.
Remark: A metric space is a metrizable space.
Def: Let X be a metric space with metric d. A subset A of X is
said to be bounded if there is some number M such that d(x, y)
 M for every pair x, y of A.
Thm: Let X be a metric space with metric d. Define d: X  X
 R by the equation d(x, y) = min {d(x, y), 1}. Then d is a
metric that induces the same topology as d.
1.8 Continuous Functions
Def: Let X and Y be topological spaces. A function f: X  Y is
said to be continuous if for each open set V of Y, the set f- -1 (V)
is an open set of X.
Prop.: Let X and Y be topological spaces, and B a base of the
topology T of Y. A function f: X  Y is continuous iff the
inverse image of every base element is open.
Proof: The arbitrary open set V of Y can be written as a union
of base elements: V =  {B :   J}, and we have that f- -1 (V)
=  {f- -1 (B ):   J},
Prop.: Let X and Y be topological spaces, and S a subbase of
the topology T of Y. A function f: X  Y is continuous iff the
inverse image of every subbase element is open.
Proof: The arbitrary base element B of Y can be written as a
finite intersection of subbase elements: B = S1  S2  …  Sn ,
and we have that f- -1 (B) = f- -1 (S1 )  f- -1 (S2 )  …  f- -1 (Sn ).
Ex.: Let T be the usual topology on R, and T l the lower limit
topology on R. Then, the identity function f: x  x is not a
continuous function from (R, T ) to (R, T l ). On the other
hand, the identity function g: x  x is a continuous function
from (R, T l ). To (R, T )
Thm.: Let X and Y be topological spaces and f: X  Y is a
function . Then, the following are equivalent:
(1)
(1)
(2)
(3)
f is continuous.
For every subset A of X, we have f- (cl(A))  cl( f (A)).
For every closed subset B of Y, the set f- -1 (B) is closed in X.
For each x  X and each nbhd. V of f (x), there is a nbhd. U
of x such that f (U)  V.
Proof: (1)  (2) Assume that x  cl(A). If V is an open nbhd. of
f (x), then f -1 (V) is an open nbhd. of x. So, it must intersect A in
some point y. It follows that V intersects f (A) in the point f (y).
This implies that f (x)  cl( f (A)).
(2)  (3) Let B be closed in Y and A = f- -1 (B). If x  cl(A),
then f (x)  f (cl(A))  cl(f (A))  cl(B) = B. Therefore,
x  f- -1 (B) =A.
(3)  (1) Let V be an open set in Y and B = Y – V. Then
f- -1 (B) = f- -1 (Y) - f- -1 (V) = X - f- -1 (V) . This means that
f- -1 (V) is open in X.
(1)  (4) Let x  X and let V be an open nbhd. of f (x). Then U
= f- -1 (V) is an open nbhd. of x, and f (U)  V.
(4)  (1) Let V be an open set in Y. For each x  f- -1 (V), we
have f (x)  V. There is an open nbhd. U of x such that f (U)  V,
This implies that U  f- -1 (V). Therefore, f- -1 (V) can be
written as the union of open sets.
Def: Let X and Y be topological spaces, f: X  Y be a bijection .
If both the function f and its inverse f- -1 : Y  X are continuous,
then f is called a homeomorphism.
Remark 1: Another way to define a homeomorphism is to say
that f: X  Y is a bijective correspondence such that f (U) is
open U is open.
Remark 2: A homeomorphism f: X  Y gives us a bijective
correspondence not only between X an Y, but between the
collection open sets of X and Y. So, any property of X that is
entirely expressed in terms of the topology of X yields the
corresponding property of Y.
Ex: A bijective function f: X  Y can be continuous without
being a homeomorphism.
Let S1 denote the unit circle, S1 = {(x, y): x2 + y 2 =1},
considered as a subspace of the plane R2 , let f : [0, 1)  be the
map defined by f (t) = (cos2t, sin2t).
(1) f is bijective and continuous.
(2) The image of the open set U = [0, 0.2)  [0, 1) under f is not
open in S1.
Local formulation of continuity: The map f: X  Y is continuous
if X can be written as the union of open sets U such that f U is
continuous for each B .
Proof: Let X be the union of open sets U and each f U is
continuous. For each open set V in Y, it is easy to check that
f- -1 (V)  U = (f  U) -1 (V). Since f U is continuous, the set
f- -1 (V)  U is open in U, and thus open in X. So,
f- -1 (V) =  (f- -1 (V)  U ) is open in X.
The pasting lemma: Let X = A  B, where A and B are closed in
X. Let f: A  Y and g: B  Y be continuous. If f (x) = g (x) for
every x  A  B, then f and g combine to give a continuous
function h: X  Y, defined by setting h (x) = f (x) if x  A, and
h (x) =g (x) if x  B.
Proof: Let C be closed subset of Y, then we have
h -1 (C) = f- -1 (C)  g -1 (C) by elementary set theory. Since f is
continuous, f- -1 (C) is closed in A, and thus closed in X.
Similarly, we can prove that g -1 (C) is closed in B, and therefore
closed in X. It follows that h -1 (C) is closed in X.
1.8 Sequential Spaces.The Sequential Closure Operator
Definition 9. A topological space X is called sequential if , for
every set A  X which is not closed in X , there exists a sequence

 xn n 1 of points of A converging to a point of the set A \A .
Sequential closure :  Aseq  x  X : sequencexn n1 in A , xn  x; A  Aseq is
a map from 2 X  2 X , it is called the sequential closure operator. It
can be defined in any topological spaces.
Properties :
0 seq
1seq
2 seq
Remark:
.  seq  
. A   Aseq  A
.  A B    A  B 
 A    A
seq
 seq  seq
seq
seq
in general.
seq
Example. Let X be the set consisting of 3 different type points:
isolated points : xmn .


limit points : yn of sequence  xmn : m  N 

point z : with most complicated local structure
The collection B =  xmn  : n  N  , m  N    Vk  yn  : n, k  N   is a base of
a topology on X , where Vk  yn    yn  xmn : m  k .
and   W : z W , p  N  ,Vk  yn  \ W is finite and yn  W for all n  p  .
1. X is sequential space (Arens Fort space, Modified) Because
only those yn ' s and z could be limit point of sequence with
distinct points . If yn ' s or z in G \ G , there must be a sequence in
G converging to it.
2. Let
A   xmn : m  N  , n  N   , B   yn : n  N  
 B seq  B  z ,therefore z   Aseq  seq
 A    A
seq
 seq  seq
, then  Aseq  A
, but z   A
. Thus
B
seq
.
Definition 10 . A topological space X is called a Frechet-Uryshon
space if the closure of every subset A  X in X coincides with the
sequential closure of A : A   Aseq .
It follows that Frechet-Uryshon space is sequential and from
the above example, it shows that not every sequential space is
Frechet-Uryshon space.
Example 14. Consider the subspace Y  A z of the space X
in example. It is easy to verify that no sequence of points in A
converges to the point z. But A A  z . Consequently, Y is not
a sequential space. Thus a subspace of a sequential space need
not be sequential.
Proposition 6. A topological space X is a Frechet-Uryshon
space if and only if each subspace Y of X is sequential.
Proof:   Let Y  X with subspace topology. A  Y , if A is
not closed in Y.Then clY A \ A   , clY A   cl X A Y   Aseq Y , For any
point x in( Aseq Y ) \ A , there is a sequence in A converging to x.
Therefore Y is sequential.
  For any subset
, let y  A \ A , consider Y  A  y , then Y

is sequential. Therefore in A  sequence  xn n 1 converging to y,
this shows that A   Aseq hence A   Aseq . X is Frechet-Uryshon
space.
A X
1.9. The First Axiom of Countability and Bases of a Space at a
Point (and at a Set)
Definition11. A collection  of open neighborhoods of a point x
in a topological space X is called a base of the space X at the point
x if each neighborhood of x contains an element of  .
First Axiom of countability: each point has a countable base. C 
Example: Space with countable base, all discrete spaces
satisfies C .
Note that: the discrete space X has countable base  X itself is
countable. Hence not every C space is C space.
Proposition 7. If X satisfies C  X is a Frechet-Uryshon space.
Proof : For any set A  X ,we show A   Aseq . Let x  A , suppose
 is a countable base at x, say   On : n    , WLOG we may
assume that Om  On when m  n , otherwise we can simply let
Om  mi 1 Oi , then replace Om by Om . It is easy to see that A Om  
for any m   , pick xm  A Om , then  xn : n    is a sequence in A
converging to x. For any neighborhood U of x, Om   such that
x  Om  U , then xn  Om  U for all n  m .
Proposition 8. A countable space X is
base.
C  it
has countable
Proof :   Trivial.
 For each point x, Let Bx be a countable neighborhood base
S  Bx : x  X  is a countable collection, use it as subbase, the
topology generated by S is the original topology T .
B  S1   Sn : Si S  is a countable base for T .
Example 15. X in Example 13, Y in Example 14 are countable
, but do not have C .They are not even Frechet-Uryson space.
Thus , not every countable space has a countable base. Note that
all single point sets (singletons) in Y are closed and only one point
(the point z) is not isolated.
1.10 Everywhere Dense Sets and Separable Space.
Definition 13. A subset A of a topological space X is called
everywhere dense in X if its closure is equal to X: A  X .
: equivalent to for every nonempty open U , U A  
: enough for every nonempty base element U B U A  
X is separable if it contains a countable dense subset.
Example 20. Every countable space is separable.
is dense in
 ,T  as well as  , T S  .  ,T  has countable base,  , T S 
does not.
Proposition 10. Not every separable space satisfying the first
axiom of countability possesses a countable base.
Proposition 11. Every space with a countable base is separable.
Proof : Choose a point in each nonempty base element B . A of
all such points is countable and dense in X .
C is hereditary property.
Note 1. Separability is not hereditary.
Example(P149):  X , T  is a topological space ,  X ,X *  X  ,
T *   A {}| A  T    ,  X * , T *  is a space.
1  X * , T *  is separable .  belongs to any nonempty open set ,
then  is a dense set in  X * , T *  .
2  X , T  can have any properties we want . For example choose
any nonseparable space , say uncountable discrete space.  X , T 
is a subspace of  X * , T *  .
Note 2. Even every subspace of X is separable(X is hereditarily
separable ). X need not be C .
Sorgenfrey line: not C but hereditarily separable. Think of how
to prove it.
Countable space without a countable base Example 14 or
Example 19 , countable Frechet-Uryshon fan.