Bases and sub bases

Introduction:

The aim of this lecture is to complete the subject of the bases and the sub bases of the topological spaces ,in the last lecture we studied and investigate some definitions of the bases and the sub bases also we discussed some theorem in this topic ,also in this lecture we shall study some properties of a topological sub spaces and the meaning of the hereditary property, many theorem and results are discussed in this day .

Theorem:
Let (X,?) be a topological space ,a sub collection ? of ? is base for ? iff every open set can be expressed as a union of members of ?.

Proof:
? let ? be a base for ? and let G be open set then for all x G ,there exists B ? such that x B G, then G= { B ?: B G}.
? let ? ? ,and let every open set G be the union of members of ?, we have to prove that ? is a base for ?,
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 ? then there exist B ? such that x B G then ? is a base for ?.

Theorem:
Let are two topologies defined on X which have a common base ? then .

Proof:
let G is open set and x G, since G is open set then it is nbd of x, since ? is a base for ? then there exist B ? such that x B G, since ? is abase for and B ? then B
hence G is open set in ,since x is arbitrary then G then or similarly ,hence .







Hereditary property( الخاصية الوراثية )

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}, ?={X,?,{1},{3,4},{1,3,4},{2,3,4,5} } and let Y={1,4,5} X then find the relative topology for Y.
? ={ ?,Y,{1},{4},{1,4},{4,5} }.


Theorem:
Let (Y, ? ) be a sub space of a topological space (X, ?) and (Z,W) be a sub space of (Y, ? ) then(Z,W) is sub space of (X, ?).



Theorem: (exercise)
Let (Y, ? ) be a sub space of a topological space (X, ?) ,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 .
ii) for every sub set A in Y we have cl (A)= cl (A) Y.
iii) for every sub set A in Y we have int (A) int (A).

Theorem:
Let (Y, ? ) be a sub space of a topological space (X, ?) and if A Y is open(closed) in X then it is also open(closed) in Y.

Theorem:
Let (Y, ? ) be a sub space of a topological space (X, ?) and let ? be a base for ? then ? ={ B Y: B ? } is abase for ? .