Lecture 1: Topology
Topological spaces and basic definitions
Introduction :
In this lecture we shall try to investigate some basic definitions in a topological spaces, so what is the topology and what is the types of topologies and what is the main concepts in topology ,this lecture will answer all these questions
Definition :
Let X be a non-empty set and let

be a collection of a subset of X satisfying the following conditions: i) X,
   ii) if
G
1
, G
2
 
,then
G
1
 G
2
  iii) 
  
G
 i
 
for
  
.i
 I then

is called a topology on X and (X,

) is a topological space and for easy written by X.
Remark: the elements of

is called open sets .
Example:
Let X={a,b,c},

={X,

,{a} }. Is topology on X, since
 satisfying the conditions (i),(ii) and (iii).
Examples:
Let X={a,b,c} ,then

1
={X,

}.
I

 2
={X,

,{a}}.
 3
={X,

,{b}}.
 4
={X,

,{c}}.
 5
={X,

,{a,b}}.
 6
={X,

,{a,c}}.
 7
={X,

,{b,c}}.
 8
={X,

,{a},{a,b}}.
 9
={X,

,{b},{a,b}}.
 10
={X,

,{c},{a,c}}.
 11
={X,

,{c},{c,b}}.
 12
={X,

,{b},{c,b}}.
 13
={X,

,{a},{a,c}}.
 14
={X,

,{a},{b},{a,b}}.
 15
={X,

,{a},{c},{a,c}}.
 16
={X,

,{c},{b},{c,b}}.
 17
={X,

,{a},{a,c} ,{a,b}}
 18
={X,

,{a},{a,c} ,{a,b}}
 19
={X,

,{c},{a,c} ,{c,b}}
 20
={X,

,{b},{b,c} ,{a,b}}.
 21
={X,

,{a},{a,c} ,{a,b},{b} }
 22
={X,

,{a},{a,c} ,{a,b},{c} }
 23
={X,

,{c},{a,c} ,{c,b},{b} }
 24
={X,

,{c},{a,c} ,{c,b},{a} }
 25
={X,

,{b},{b,c} ,{a,b},{a} }.
 26
={X,

,{b},{b,c} ,{a,b},{c} }.
 27  exercise.
 28
= exercise
 29
={X,

,{b},{a},{c},{a,c},{b,c} ,{a,b}}.
Remark:

1
={X,

}.is called an indiscrete topology .
II

 29
={X,

,{b},{a},{c},{a,c},{b,c} ,{a,b}}. Is called a discrete topology .
Example: letX={a,b,c},

={X,

,{a,c} ,{c,b},{a} },then
 is not topology on X, since {a,c}

{c,b}={c}
 
Example: letX={a,b,c},

={X,

,{a,c} ,{c,b},{b} },then

is not topology on X, since {a,c}

{c,b}={c}
 
Example: letX={a,b,c},

={X,

,{a,c} ,{a,b},{c} },then
 is not topology on X, since {a,c}

{a,b}={a}
 
Example: let X={a,b,c},

={X,

,{a,c} ,{c,b},{a},{b}
},then

is not topology on X, since
{a,c}

{c,b}={c}
  and {a}

{b}={a,b}
 
Example: let X={a,b,c},

={X, Ø,{a,c} ,{c,b},{a},{b}
},then

is not topology on X, since
  
.
Exercise: let X={1,2,3},then find a topologies on X.
Example: let X=N, where N is the set of natural numbers then
,C={G

N:X/G is finite} is a topology on X ,and it is called co finite topology prove that C is a topology on X=N.
Also if V={G

N:X/G is countable} is a topology on X and it is called co countable topology on X ( prove that ).
Example: let U consist of Ø and all those subset G of R having the property that to each x X G there exist X >0
III

such that (xX , x+ X ) T G show that U is a topology for
R called the usual topology .
Exercise: let X={a,b,c,d},then find a topologies on X.
Theorem: the intersection of a family of topologies defined on the same set is a topology on this set. remark: the union of a family of topologies defined on the same set need not be a topology on this set
Example: let X={a,b,c},

1
={X,

,{a}} and
 2
={X,

,{b}}are two topologies on X but
 
1

2
={
X,

,{a},{b}} is not topology on X .
Definition:
Let (X,T) be atopological space and let x X X , a subset
Nof X is said to be a T-neighbourhood of x iff there exist a T-open set G such that x X G T N.
Example:
X={1,2,3,4,5}
T={X,  , {1},{1,2},{1,2,5},{1,3,4},{1,2,3,4}}
T-nbd of 3 are
{1,3,4}, {1,2,3,4},{1,3,4,5} and X
Find nbd of 1 and 5
Theorem: a subset of a topological space X is open iff it’s a nbd of each of its point
Proof : discussed in the lecture
Theorem:
IV

Let X be a topological space and for each x X X , let N(x) be the collection of all nbds of x then:
1 x X X , N(x)≠Ø .
2N X N(x)then x X N.
3N X N(x) , N T M then M X N(x).
V
4N X N(x) , M X N(x) then N W M X N(x).
Proof: discussed in the lecture
Definition:
Let 
1
, 
2
be two topological spaces defined on the same non-empty set X ,we say that 
1 is finer than 
2
or 
2
is coarser than 
1
if 
2
 
1
.
Definition:
Let 
1
, 
2
be two topological spaces defined on the same non-empty set X ,we say that 
1
and 
2
are no comparable if neither 
2
 
1
nor 
1
 
2
.
Example:
Let X={a,b,c,d} ,and 
1
={  ,X,{a},{a,b},{a,b,c} } and

2
={  ,X,{a},{a,b} } then 
1
is finer than 
2
since 
2
 
1
.

Example:
Let X={a,b,c } and 
1
={  ,X,{a},{a,b} } ,

2
={{  ,X,{a} } ,is 
1 is finer than 
2
?
Definition:
Let (X, τ) be a topological space and Y  X , we can define a topology on Y by the intersection of Y with each open sets with Y. and it is defined by:

Y
={ Y  G : G   }.
Example:
Let X={a,b,c } and τ={ X,{a},{b}, Φ,{a,b} } and
Y={a,c} then find a topology on Y.
Solution:

Y
={ Y  G : G   },then 
Y
={ Y, Φ, {a} }.
Exercise:
let X={a,b,c} and Y={b,c} and
τ ={X,  ,{c},{c,b}}.
τ ={X,  ,{b},{c,b}}.
τ ={X,  ,{a},{a,c}}.
τ ={X,  ,{a},{b},{a,b}}.
τ ={X,  ,{a},{c},{a,c}}.
τ ={X,  ,{c},{b},{c,b}}.
τ ={X,  ,{a},{a,c} ,{a,b}}
VI

τ ={X,  ,{c},{a,c} ,{c,b}}
τ ={X,  ,{b},{b,c} ,{a,b}}.
τ ={X,  ,{a},{a,c} ,{a,b},{b} }
τ ={X,  ,{a},{a,c} ,{a,b},{c} }
τ ={X,  ,{c},{a,c} ,{c,b},{b} }, then find the topologies on Y.
VII