Question

Al. Let E be a non-empty set and let d:ExE0, oo). (a) Give the three conditions that d must satisfy to be a metric on E. (b) Ifa E E, r > 0 and 8 0, give the definition of the open ball BE(a) and the closed ball B (a) n-p) closure point of A. Hence, say what it means for A to be a closed subset of E 2 c) Say what it means for a sequence () in E to converge to a point pE E (denoted by (d) Let A be a subset of a metric space E. Say what it means for a point p E E to be a (e) Show that if p, q E E and zn → p, xn → q, then p = q. 2

Answer 1

(a) Three condition are following

(1) d(x,y) = 0 if and only if x=y. (2) d(x,y) = d(y,x) for all x,y $\in E$ (3) d(x,y) $\leq$ d(x,z) + d(z,y) for all x,y and z $\in E$ .

(b) Open ball of radius r around a = $\{x\in E: d(a,x)<r\}$ . Closed ball of radius r around a = $\{x\in E: d(a,x)\leq r\}$ .

(c) A sequence ( $x_n$ ) converges to a point p if for every  open neighbourhood U of p  there exists $k \in \mathbb{N}}$ such that such that $x_n$ $\in U \;\textnormal{for all}\; n \geq k$ .

(d) A point $p \in E$ is closure point of A if every neighbourhood of p intersect with A.

(e) Using the third condition of metric space and definition of convergence one can easily proof the required result.