**Definition**: Let* X *be a non empty set. *A* class T of subsets of *X* is a topology on *X* if and only if *T* satisfies the following axioms.

*O*

_{1}] X and ∅ belong to

*T*. [

*O*

_{2}] The union of any number of sets in

*T*belongs to

*T*. [

*O*

_{3}] The intersection of any two sets in

*T*belongs to

*T*.

The members of *T* are then called *T*– open sets or simply open sets, and *X* and *T* together i.e., the pair (*X, T*) is called a topological space.

# Example: Consider the following classes of subsets of

*X* = {*a, b, c, d, e*}

*T*_{1} = {*X*, ∅, {*a*}, {*c, d*}, {*a, c, d*}, {*b, c, d, e*}}

*T*_{2} = {*X*, ∅, {*a*}, {*c, d*}, {*a, c, d*},{*b, c, d*}}

*T*_{3} = {*X*, ∅, {*a*}, {*c, d*}, {*a, c, d*}, {*a, b, d, e*}}

Observe that *T*_{1} is a topology on *X* since it satisfies the necessary three axioms [*O*_{1}], [*O*_{2}], [*O*_{3}]. But *T*_{2} is not topology on *X* since the union of two members of *T*_{2} does not belong to *T*_{2}, i.e.,*T*_{2} does not satisfy the axiom [*O*_{2}].

{*a, c, d*} ⋃ {*b, c, d*} = {*a, b, c, d*}

Also, T_{3} is not a topology on X since the intersection of two sets in T_{3} does not belong to T_{3}, i.e., T_{3} does not satisfy the axiom [O_{3}].

{*a, c, d*} ⋂ {*a, b, d, e*} = {*a, d*}

**Co-finite topology**: Let *T* denote the class of all subsets of *X* whose complements are finite together with the empty set ∅. This class* T* is also a topology on *X*. It is called the Co-finite or the *T*_{1}– topology on *X.*

**Coarser and Finer topologies**

Let T_{1} and T_{2} be topologies on a non-empty set X. Suppose that each T_{1} – open subset of X is also a T_{2}– open subset of X. That is, suppose that T_{1} is a subclass of T_{2}, i.e., T_{1} ⊂ T_{2}. Then we say that T_{1} is coarser or smaller than T_{2} or that T_{2} is finer or larger than T_{1}.

**Subspaces, Relative topologies**

Let A be a non empty subset of a topological space (X, T). The class T_{A} of all intersections of A with T-open subsets of X is a topology on A; it is called the relative topology on A or the relativization of T to A, and the topological space (A, T_{A}) is called a subspace of (X, T).

Example: Consider the topology

T = {*X*, ∅, {*a*}, {*c, d*}, {*a, c, d*}, {*b, c, d, e*}}

On *X* = {*a, b, c, d, e*}, and the subset A = {a, d, e} of X. Observe that

*X* ⋂ *A* = *A*, {*a*} ⋂ *A* = {*a*}, {*a, c, d*} ⋂* A *= {*a, d*}, ∅ ⋂ *A *= ∅, {*c, d*} ⋂ *A* = {*d*}, {*b, c, d, e*} ⋂ *A* = {*d, e*}

Hence the relativization of T to A is

*T _{A}* = {

*A*, ∅, {

*d*}, {

*a, d*}, {

*d, e*}}

**Theorem: Let ( X, T) be a subspace of (Y, T^{*} ) and let (Y, T^{*} ) be a subspace of (Z, T^{**}) . Show that (X, T) is also a subspace of (Z, T^{**}).**

Proof: Since *X* ⊂ *Y* ⊂ *Z*, (*X, T*) is a subspace of (*Z, T ^{**}*) if and only if

*T*=

_{x}^{**}*T*. Let

*G*

*∊*

*T*; now

*T*=

_{x}^{*}*T*, so there exist

*G*

^{*}*∊*

*T*

_{x}^{*}for which

*G = X*

*⋂*

*G*But

^{*}.*T*so there exist

^{*}= T_{y}^{**},*G*∊

^{**}*T*such that

^{**}*G*=

^{*}*Y*⋂

*G*Thus

^{**}.*G* = *X* ⋂ *G ^{*} *=

*X*⋂

*Y*⋂

*G*=

^{**}*X*⋂

*G*

^{**}since *X*⊂ *Y*; so *G* ∊ *T ^{**}. *Accordingly,

*T*⊂

*T*

_{x}^{**}.Now assume *G* ∊ *T _{x}^{**}, *i.e. there exist

*H*∊

*T*

^{*}^{*}such that

*G*=

*X*⋂

*H*.

But *Y* ⋂ *H* ∊ *T _{y}^{**} *=

*T*so

^{*}*X*⋂ (

*Y*⋂

*H*) ∊

*T*=

_{x}^{*}*T*. Since

*X*⋂ (

*Y*⋂

*H*) =

*X*⋂

*H*=

*G*

We have *G* ∊ *T*. Accordingly, *T _{x}*

^{**}⊂

*T*and the theorem is proved.