Basis Topology - Topology, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

1. Defining a Topolgy

Let X be a set and τ a subset of the power set of X . Then the a pair (X, τ) is said to define a topology on a the set X if τ satisfies the following properties :

(1) If φ and X is an element of τ .
(2) If union of any arbitrary number of elements of τ is also an element of τ.
(3) If intersection of a finite number of elements of τ is also an element of τ.

Now we define the following terms which arise as consequence of the above construction :

(1) We call the elements of τ as ’open sets’.
(2) We define all other subsets of X which are not contained in τ as “closed sets”.
(3) After the set has been endowed with a topology we call X as the “ underlying set” for the topology .
(4) Earlier what were called the elements of the set X will be called points of the topological space (X, τ) once a topology has been defined on X .

The motivation behind restricting the intersections to finitely many is that if arbitrary number is allowed then the inetersection of all intervals in being a single point that is 0 will be declared open! . This goes against our intuition about real numbers and hence this has been prevented by inserting the finiteness condition.

2. Base for the topology

We give here two definitions for the base for a topology (X, τ).

(1) We call a subset B1 of τ as the “Base for the topology” if every set in τ can be obtained by union of some elements of B .

(2) We call a subset B2 of τ as the “Basis for the topology” if for every point x ∈ U ⊂ τ there exists an element of B2 which contains x and is a subset of U .

We now need to show that B1 = B2.

B1 ⊂ B2. First we show that B1 ⊂ B2

Let U be an open set under the topological structure considered . By definition 1 there exists a subset of B1 such that their union is equal to U . Hence every element of B1 which participates in this union is a subset of U .

Further every point in U is contained in atleast one of the sets of those who have participated in the union and by the last sentence each of them is a subset of U .

Finally since every point of X is contained in atleast one element of τ (lets call that U) (trivially X is always there!)and since that element of τ can be written as a union of elements on B1 by the first two sentences there exists an element of B1 containing that point and is a subset of U

Hence every element of B1 which contributes in the union to create U is an element of B2.Hence after scanning through all elements of τ to access all the elemnts of B1 we can see that B1 ⊂ B.

B2 ⊂ B1. This is fairly simple. We look at at each point x ∈ U and by definition there exists atleast one element of B2 which contains x and is a subset of U .So around every point in U we can find an element of B2 which is completely contained in U . So we take the union of all such elements of B2 and that will exactly be equal to U. Hence the elements of B2 which contribute fopr this U satisfy the condition of being in B1 with respect to the open set U

Since this can be done for every U and by doing this for every U ⊂ τ we can access all the elements of B2 .Thus defining criteria of B1 is collectively satisfied and hence B2 ⊂ B1.

Since B1 ⊂ B2 and B2 ⊂ B1 , we can conclude that B1 = B2.

Hence the basis generated by the two definitions are equal.

3. Creating a topology from a given base on a set

Base of a set. We now have just a set X and we define that B3 (a subset of the power set of X) will be said to be a base for X if :

(1) If for every element x of X there exists a element of B3 containing it .

(2) If P and Q are 2 elements of B3 which contain the point x then there exists a another element R of B3 which contains x and is a subset of P ∩ Q.

Giving a topology on the set X through its base (as defined above). Now given this way of defining a “base for a set” , there is no obvious way a topology can be endowed on X but there exists a natural way as follows :

Declare a subset U of the power set of X to be “open” if for every point x ∈ U there exists a B ⊂ B3 such that x ∈ B and B ⊂ U .

Now the set of all such U as defined above will be the topology τ on X and (X, τ) forms the topological space.

Consistency check. That φ is an element of τ is vacuously true.

That X ⊂ τ is also obvious as the first criteria of a “base” guarantees existence of an element of B3 containing every point of X and all such elements of B3 are obviously subsets of X . Hence X is open

Let P and Q be elements of τ and hence there exists an element of B3 for every element x ∈ P ∪ Q which contains x and is is a subset of P ∪ Q since it is a subset of either P or Q (by definition). Hence P ∪ Q is an element of τ .This can be iterated arbitary number of times and hence arbitary unions are also elements of τ

By definition of a “base of a set” for every element x in P ∩ Q there exists atleast one element U in B3 such that x ∈ U and U ⊂ P ∩ Q . Hence P ∩ Q qualifies to be in τ.

Now we do induction on collection of n open sets indexed by i like U_i .

i = 1 is trivially true and let us aaume that it is true till i = n − 1. let J be the intersection till n − 1 . Then the case of the intersection of J and Un is the same as the case for 2 open sets as shown earlier. Hence induction folows.

Hence the collection of open sets defined above indeed satisfy the conditions to form a topology over X .

4. Lets reverse the above process!

Now given a topological space (X, τ) (as derived from the above construction) we create a base for it by using the definition of the last section that is :

B3 (a subset of the power set of X) will be said to define a base for X if :

(1) If for every element x of X there exists a element of B3 containing it .

(2) If P and Q are 2 elements of B3 which contain the point x then there exists a another element R of B3 which contains x and is a subset of P ∩ Q.

B3 ⊂ B2. Let us take an openset U in τ and an arbitary point x in it . So by definition of B3 since x ∈ X there exists some B ∈ B3 such that x ∈ B . Since these open sets are defined according to the construction that follows from B3 in the last section we can find a B ∈ B3 that not only contains x but also is a subset of U . So by scanning through all opensets in τ we have all the elements of B3 and each of them qualifies to be an element of B2.
Hence B3 ⊂ B2.

B 2 ⊂ B 3. This is slightly tricky! We again take an openset U ⊂ τ and there exists an element B of B 2 such that x ∈ B ⊂ U .lrt C be another element in B 2 which contains x. By definition C ⊂ U . Therefore B ∩ C is also a subset of U and it contains x . Now think of B ∩ C as another set which is a subset of τ . So by re-invoking the definition of B 2 we are again guaranteed that there exists an element D ⊂ B 2 which contains x and is a subset of B ∩ C .

Hence the triplet B , C and D together satisfy the conditions to be in B2.

Hence by scanning all the opensets we can track down all the elements of B 2 and in triplets they all are qualifying to be elements of B3. Hence B2 ⊂ B3.

Since B2 ⊂ B3 and B3 ⊂ B2 we have that B2 = B3 . Hence we have that the two ways of giving a base to the same base to the topology.

5. A slight digression

6. Neighbourhood

Let (X, τ) be a topological space. Let x ∈ X be a point in this space.

Open Neighbourhood. An openset U ⊂ τ containing x is called an “open neighbourhood” of x .

Neighbourhood. A set (element of the power set of X) is called a “neighbourhood” of x if it contains an “openset” containing x alternatively contains an open neighbourhood containing x.

Open Cover. An Open Cover of X is a family of opensets {U_α} such that each U_α is an element of τ and  

Compactness. A topological space (X, τ) is called “Compact” if from every open cover of it a finite subcover can be extracted.

Local compactness. A topological space (X, τ) is called “locally compact” if every point in this space has a compact neibhbourhood.

Fundamental system of Neighbourhoods. A set of neighbourhoods of a point x ,{U_s}{s ∈ S ⊂ I} is called a fundamental system of neighbourhoods of x if every neighbourhood of x contains atleast one of the U_s.

The important claim. The claim is that Compact Neighbourhoods of a point form a fundamental system of neighbourhoods for the point

7. Conclusion

Hence we have shown that the 2 ways of giving a base to a topology are the same as they give the same base and that the opensets obtained natural ly from the process of giving a “base to the underlying set” indeed gives a topology to the set . Further we have shown that if one of the earlier methods of giving a base to it is repeated we get back the “base on the underlying set” that we had started with.