... then ϕ is bounded and attains its sup and inf.Indeed, since S is compact, so is ϕ(S) and so ϕ(S)isclosed and bounded. Thus (see Exercise 1. 1-4 ) theinf and sup of this set are finite and are members ... metric is de ned as follows.1.2.2 De nition. For ε>0 and m ∈ M , the open ε-bal l (or disk) about m is de ned byDε(m)={ m∈ M | d(m,m) <ε} , and the closed ε−bal l is de ned byBε(m)={ ... setscontained in A. The boundary of A, denoted bd(A) is de ned bybd(A)=cl(A) ∩ cl(S\A).By C3, cl(A)isclosed and by T3,int(A)isopen. Note that as bd(A)isthe intersection of closed sets,bd(A)isclosed,...