If A is a subset of a topological space, then ∂(A)⊆Cl(A).

Proof:

To prove that If A is a subset of a topological space, then ∂(A)⊆Cl(A). OR Let A be a subset of a topological space (X, T). Then ∂(A)⊆cl(A).

Related Theorems;

Show finite complement topology is, in fact, a topology.

Show that the intersection of two topologies on X is a topology.

We have need to focus on definitions of boundary and Closure of Subset A of topological Space (X, T) to show Bd(A) ⊆ Cl(A).

1) Let    a ∈ ∂(A) 

Then by definition of boundary of set A where A is the subset of topological Space (X, T), we have

a ∈ A or a ∈ Cl(A), but a ∉ Int (A).

2) If a ∈ A

Then by definition of closure of set A where A is the subset of topological Space (X, T),

we have  a ∈ Cl(A) 

Since by definition of boundary of subset of topological Space,

If a ∈ Cl(A), then a ∉ Int (A).

In either case, a ∈ cl(A). Therefore, ∂(A) ⊆ Cl(A).

Hence proved that If A is a subset of a topological space, then ∂(A)⊆Cl(A).