33 C
Maharashtra
Friday, June 9, 2023
HomeConnectednessAny two closed subset of metric space are connected iff they are...

Any two closed subset of metric space are connected iff they are disjoint

- Advertisement -

Theorem:

Any two closed subset of metric space are connected iff they are disjoint.

Proof:

Let (X, d) be a metric space and A & B are any two closed subsets of X.
Let A & B are separated sets.
Claim: A & B are disjoint.
As A & B are separated, 
\( \bar{A}\cap B=\phi \)
\( \therefore A\cap B=\phi \)​ (​\( \because \)​ A is closed ​\( \implies A=\bar{A} \)​) 
Conversely,
Suppose that A & B are disjoint.
Claim: A & B are separated.
By hypothesis,
\( A\cap B=\phi \)
Since, A is closed, ​\( A=\bar{A} \)
\( \implies \bar{A}\cap B=\phi \)​ ———–(1)
Similarly, B is closed, ​\( B=\bar{B} \)
\( \implies A\cap \bar{B}=\phi \)​ ———–(2)
From (1) & (2),
A and B are separated sets.
- Advertisement -
RELATED ARTICLES

LEAVE A REPLY

Please enter your comment!
Please enter your name here

- Advertisment -

Most Popular

Insert math as
Block
Inline
Additional settings
Formula color
Text color
#333333
Type math using LaTeX
Preview
\({}\)
Nothing to preview
Insert