# Let (X,τ) be a topological space ,a sub collection β... every open set can be expressed as a union of... Theorem:

```Theorem:
Let (X,τ) be a topological space ,a sub collection β of τ is base for τ iff
every open set can be expressed as a union of members of β.
Proof:
→ let β be a base for τ and let G be open set then for all x  G ,there
exists B  β such that x  B  G, then G=  { B  τ: B  G}.
← let β  τ ,and let every open set G be the union of members of τ, we
have to prove that β is a base for τ,
Let x  X, and let N be a nbd of x, then there exists an open set G such
that x  G  N ,but G is the union of members of β then there exist B  β
such that x  B  G then β is a base for τ.
Theorem:
Let  1 , 2 are two topologies defined on X which have a common base β
then  1   2 .
Proof:
let G is open set and x  G, since G is open set then it is nbd of x, since β
is a base for τ 1 then there exist B  β such that x  B  G, since β is abase
for  2 and B β then B   2
hence G is open set in  2 ,since x is arbitrary then G   2 then  1   2 or
similarly  1   2 ,hence  1   2 .
Hereditary property( ‫) اﻟﺨﺎﺻﯿﺔ اﻟﻮراﺛﯿﺔ‬
Definition:
A property of a topological space is said to be hereditary property if every
sub space of the space has that property.
Example:
Let X={1,2,3,4,5}, τ={X,Φ,{1},{3,4},{1,3,4},{2,3,4,5} } and let
Y={1,4,5}  X then find the relative topology for Y.
I
τ Y ={ Φ,Y,{1},{4},{1,4},{4,5} }.
Theorem:
Let (Y, τ Y ) be a sub space of a topological space (X, τ) and (Z,W) be a
sub space of (Y, τ Y ) then(Z,W) is sub space of (X, τ).
Theorem: (exercise)
Let (Y, τ Y ) be a sub space of a topological space (X, τ) ,then
i)
a sub set A of Y is closed in Y iff there exist a closed set F in
X such that A=F  Y .
ii)
for every sub set A in Y we have cl Y (A)= cl X (A)  Y.
iii) for every sub set A in Y we have int X (A)  int Y (A).
Theorem:
Let (Y, τ Y ) be a sub space of a topological space (X, τ) and if A  Y is
open(closed) in X then it is also open(closed) in Y.
Theorem:
Let (Y, τ Y ) be a sub space of a topological space (X, τ) and let β be a base
for τ then β Y ={ B  Y: B  β } is abase for τ Y .
II
```