Friday, February 3, 2023
Google search engine
HomeConnectednessA metric space (X, d) is disconnected iff there exists a non-empty...

A metric space (X, d) is disconnected iff there exists a non-empty proper open & closed subset of X which is both open and closed

Theorem :

A metric space (X, d) is disconnected iff there exist a non-empty proper subset of X which is both open and closed.

Proof :

Let (X, d) be a metric space and suppose that it is disconnected.
Claim: ​\( \exists \)​ a non-empty proper subset of X which is both open & closed.
Since, X is disconnected by definition, ​\( \exists \)​ non-empty sets A & B such that ​\( X=A\cup B, \bar{A}\cap B=\phi, \bar{B}\cap A=\phi \)​.
Since, ​\( A\neq\phi \)​,​\( B\neq\phi \)​ and ​\( X=A\cup B \)​, ​\( A\cap B=\phi \)​,
\( \implies A=X\backslash B \)​.
\( \therefore \)​ A is non-empty proper subset of X.
Also, ​\( B=(\bar{A})^C \)​, ​\( A=(\bar{B})^C \)​.
Clearly, A and B are open subset of X.
(Since, complement of closed set is open)
\( \therefore \)​ ​\( A=X\setminus B \)​ is closed subset of X.
Thus, A is closed as well as open subset of X.
i.e. ​\( \exists \)​ a non-empty proper set A of X which is both open and closed.
Conversely,
Suppose that metric space (X, d) has non-empty proper subset which is both open and closed.
Claim: X is disconnected.
Suppose A is non-empty proper subset of X which is both open and closed.
Let ​\( B=X\setminus A \)
\( \therefore B\neq\phi \)​, ​\( X=A\cup B \)​ and ​\( A\cap B=\phi \)
Since, A is closed and open subset of X, B is open as well as closed.
\( \therefore \bar{A}=A \)​ & ​\( \bar{B}=B \)
\( \therefore \bar{A}\cap B=A\cap B=\phi \)​ 
    ​\( A\cap \bar{B}=A\cap B=\phi \)
Thus, ​\( X=A\cup B \)​, ​\( A\neq\phi \)​, ​\( B\neq\phi \)​, ​\( \bar{A}\cap B=\phi \)​, ​\( \bar{B}\cap A=\phi \)
\( \therefore \)​ X is disconnected.
RELATED ARTICLES

LEAVE A REPLY

Please enter your comment!
Please enter your name here

- Advertisment -
Google search engine

Most Popular

Recent Comments

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