show that each open ball is open and each closed ball is clo

show that each open ball is open and each closed ball is closed

Solution

consider an open ball

by the defination of open ball

let (X,d) be a metric space if x₀  is a point of X and r is a positive real number the open ball

S_r(x₀) with centre x₀ and radious r is the subset of X and is defined by

S_r(x₀)= {x/ d(x,x₀) > r}

Open set:

A subset G of a metricspace X is called an open set for given any point X in G there exist a positive real number r such that S_r(x) contains in G

i.efor each point of G is the centre of some open ball contained in G

so, our open ball already contained in G

so every open ball is a open set

Closed ball:

If x₀  is a point in metric space X and r is a non negative real number the closed ball

S_r(x_o) ={ x / d(x,x_o)<= r}

closed set:

let X be a metric space a subset A of x is called closed iff its compliment is open

we have closed ball then

(S_r(x_o))^c ={ x / d(x,x_o)>r}

so by def of open set (S_r(x_o))^c  is open set

then S_r(x_o) is closed

then every closed ball is clossed