**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.

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

2^{3}=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)

- 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).

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