metric space

[penelope]

Show that for any metric space X , the set X \ {x} is open in X .i know that an open set in X is an open ball but i don't really get it

 

[sssshukla112]

Show that for any metric space X , the set X \ {x} is open in X .i know that an open set in X is an open ball but i don't really get it

since {x} is closed set (becoz closure of {x} ={x})
therefore its complement that is X\{x} is open set......

 

[pka]

Show that for any metric space X , the set X \ {x} is open in X .i know that an open set in X is an open ball but i don't really get it

If \(\displaystyle y\in X\setminus\{x\}\) then \(\displaystyle y\ne x\) so \(\displaystyle d(x,y)>0~.\)

\(\displaystyle (\forall y\in X\setminus\{x\})\) define \(\displaystyle \delta_y=\frac{d(x,y)}{2}\)

\(\displaystyle X\setminus\{x\}=\bigcup\limits_{y \in X\backslash \left\{ x \right\}} {B(y;\delta _y )} \).

The union of open sets is open.

 

[HallsofIvy]

Show that for any metric space X , the set X \ {x} is open in X .i know that an open set in X is an open ball but i don't really get it

Unfortunately, what you "know" isn't true. In a metric space, every open ball is an open set, but the other way is not true- not every open set is an open ball.

Let p be in X\ {x}. Let \(\displaystyle d= \delta (p, x)\). Show that the ball, centered on p, with radius \(\displaystyle \delta/2\) is in X\{x}.