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Topological Spaces | Topology

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.

[O1]   X and ∅ belong to T.

[O2]  The union of any number of sets in T belongs to T.

[O3]  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}

T1 = {X, ∅, {a}, {c, d}, {a, c, d}, {b, c, d, e}}

T2 = {X, ∅, {a}, {c, d}, {a, c, d},{b, c, d}}

T3 = {X, ∅, {a}, {c, d}, {a, c, d}, {a, b, d, e}}

Observe that T1 is a topology on X since it satisfies the necessary three axioms [O1], [O2], [O3]. But T2 is not topology on X since the union of two members of T2 does not belong to T2, i.e.,T2 does not satisfy the axiom [O2].

{a, c, d} ⋃ {b, c, d} = {a, b, c, d}

Also, T3 is not a topology on X since the intersection of two sets in T3 does not belong to T3, i.e., T3 does not satisfy the axiom [O3].

{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 T1– topology on X.

 

Coarser and Finer topologies

Let T1 and T2 be topologies on a non-empty set X. Suppose that each T1 – open subset of X is also a T2– open subset of X. That is, suppose that T1 is a subclass of T2, i.e., T1 ⊂ T2. Then we say that T1 is coarser or smaller than T2 or that T2 is finer or larger than T1.

Subspaces, Relative topologies

Let A be a non empty subset of a topological space (X, T). The class TA 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, TA) 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

XA = 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

TA = {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 XYZ, (X, T) is a subspace of (Z, T**) if and only if Tx** = T. Let G T; now Tx* = T, so there exist G* Tx* for which G = X G*. But T* = Ty**, so there exist G** T** such that G* = Y G**. Thus

G = XG* = XYG** = X G**

since XY; so GT**. Accordingly, TTx**.

Now assume GTx**, i.e. there exist H T** such that G = XH.

But YHTy** = T* so X ⋂ (YH) ∊ Tx* = T. Since X ⋂ ( YH) = X H = G

We have GT. Accordingly, Tx**T and the theorem is proved.

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