) القواعد و القواعد الجزءيه) Bases and sub bases
Introduction:
In the last lecture we studied and defined a topological spaces and them types, also we studied and investigate many facts about these spaces, also we referred to the derived sets and we knew that the family of open sets is enough to build the topology .
In this lecture we shall know by studding a family smallest than the family of the open sets which it is enough to define a topology on the set, for example ?={?,X ,{a},{b},{a,b} } on X={a,b,c}is determined by know the elements which is contained in it ?,X ,{a},{b},but this topology can be determined without {a,b},since a topology will contained it ,and by the same method the discrete topology can be determined, all this can be lead us to try know the enough sub set to determined a topology on set and this is the goal of this lecture.
The bases
Definition:
Let (X, ?) be a topological space and is called base for ? if and only if any open set G ? is a union of members of .
Remark:
A definition above is equivalent that is abase for ? if and only if for all p G such that G ?, there exist B such that p B .
Example:
Let X={a,b,c,d} and ={ {a,b},{b,c } } and let is abase for a topology ? on X then must be that the elements {a,b},{b,c} in ? so we must have {a,b} {b,c}={b} ? but {b} can not be expressed as a union of elments of ,this help us to know that the choice of must have this condition:
If A,B implies that A B= {B:B }.
Let be a family of a sub set of X ,then is a base for a topology on X if and only if the following conditions are hold:
i) X= {B:B },
ii) For any , we can expressed a bout as a union of members of .
Definition:
Let (X, ?) be a topological space and S ? is said to be sub base for ?
If and only if the finite intersection of members of S is a base for ?.
Example:
Let X={a,b,c,d,e} and S={X,{a,b,c},{c,d},{d,e} }then the family of all the intersections of members of S is ={ X,{a,b,c},{c,d},{d,e},{c},{d}, ? }, and the family of union of members of gives the topology ? = {X,{a,b,c},{c,d},{d,e},{c},{d}, ?,{c,d,e},{a,b,c,d} }.
Example:
Let X={a,b,c} and S={X,{a},{b} },then find a topology on X.
Solution:
={ X,{a},{b}, ? } and ?={ X,{a},{b}, ?,{a,b} }.
Example:
Let X={a,b,c} and S={X,{a}} ,then = ?={ X,{a}, ? }.
Example:
Let X={1,2,3} and S={X, {1},{2,3},{2} }, then ={ X, {1},{2,3},{2} ?}, ?={ X, {1},{2,3},{2}, ?,{1,2} }.
exercises:
Let X={a,b,c} and if
i) S={X,{b} } then find a topology on X.
ii) S={ X,{a},{c} } then find a topology on X.
iii) S={X, {a},{b,c},{b} } then find a topology on X.
Coarser and finer topologies( توبولوجي انعم واخشن )
Definition:
Let be two topological spaces defined on the same non-empty set X ,we say that is finer than or is coarser than if .
Definition:
Let be two topological spaces defined on the same non-empty set X ,we say that and are no comparable if neither nor .
Example:
The discrete topology is finer topology defined on a non-empty X and the indiscrete topology is coarser topology defined on a non-empty X.
Example:
Let X={a,b,c,d} ,and ={ ,X,{a},{a,b},{a,b,c} } and ={ ,X,{a},{a,b} } then is finer than since .
Example:
Let X={a,b,c } and ={ ,X,{a},{a,b} } , ={{ ,X,{a} } ,is is finer than ?
Exercise:
Let X={a,b,c } and
? ={X, }.
? ={X, ,{a}}.
? ={X, ,{b}}.
? ={X, ,{c}}.
? ={X, ,{a,b}}.
? ={X, ,{a,c}}.
? ={X, ,{b,c}}.
? ={X, ,{a},{a,b}}.
? ={X, ,{b},{a,b}}.
? ={X, ,{c},{a,c}},then determine the relation between these spaces.
Topological sub spaces( الفضاءات التوبولوجيه الجزءيه )
Definition:
Let (X, ?) 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 : G }.
Example:
Let X={a,b,c } and ?={ X,{a},{b}, ?,{a,b} } and Y={a,c} then find a topology on Y.
Solution:
={ Y : G },then ={ Y, ?, {a} }.
Exercise:
let X={a,b,c} and Y={b,c} and
? ={X, ,{c},{c,b}}.
? ={X, ,{b},{c,b}}.
? ={X, ,{a},{a,c}}.
? ={X, ,{a},{b},{a,b}}.
? ={X, ,{a},{c},{a,c}}.
? ={X, ,{c},{b},{c,b}}.
? ={X, ,{a},{a,c} ,{a,b}}
? ={X, ,{c},{a,c} ,{c,b}}
? ={X, ,{b},{b,c} ,{a,b}}.
? ={X, ,{a},{a,c} ,{a,b},{b} }
? ={X, ,{a},{a,c} ,{a,b},{c} }
? ={X, ,{c},{a,c} ,{c,b},{b} }, then find the topologies on Y.