Discrete and Indiscrete Topology

Discrete Topology

If X is a non-empty set and τ =P(X) (contains all subsets of set X), then is a discrete topology on X.it is the finer topology on set X.

Example

Let X = {a, b, c}

Possible subsets of set X.

2ⁿ (n is the total number of elements in set X)

2³=2×2×2 =8 (8 possible subset)

P(X)= {φ, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, X}

Discrete topology on set X is

τ= {φ, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, X}

Indiscrete Topology

Let X be a non-empty set and τ= {φ, x} (the set containing only the empty subset of X and set X itself) then τ is a topology on X called indiscrete topology on X.it is the weaker or coarser topology on set X.

Example

Let X = {a, b, c}

Indiscrete topology on set X is

τ= {φ, X}

Theorem

The subspace of any indiscrete topological space is indiscrete.

Proof

Let (X, τ) be an indiscrete topological space.

⟹   τ= {φ, X}

Let Y be any non-empty subset of X. We want to show that (Y, τ_y) is indiscrete topological space for τ_y. For this, we take the intersection of every number of τ with Y.

      Y∩ φ=φ

      Y∩X=Y

⟹   τ_y= {φ, Y}

⟹   τ_y is an indiscrete topology on Y.

⟹ (Y, τ_y) is an indiscrete topological subspace of (X, τ).

Theorem

The subspace of any discrete topological space is discrete.

Proof

Let (X, τ) be a discrete topological space.

⟹   τ= P(X)

Let Y be any non-empty proper subset of X. we want to show that (Y, τ_y) is a discrete topological space for τ_y.

We want to show that τ_y= P(Y)

1. For τ_y ⊆ P(Y)

As τ_y is the topology on Y.

⟹ τ_y has subsets of Y inside.

⟹ τ_y ⊆ P(Y) ——(1)

• For P(Y) ⊆ τ_y

Let B∉ τ_y

⟹B≠Y∩A ∈ τ_y    ∀ A⊆X

⟹B⊆Y

⟹B ∉ P(Y)

Hence P(Y) ⊆ τ_y ——(2)

Combining equations (1) and (2), we get

              τ_y= P(Y)

⟹ τ_y is the discrete topology

⟹ (Y, τ_y) is a discrete topological subspace of (X, τ).