Q11. Basic Topology (Rudin)
For x∈R1 and y∈R1 define
(a) d1(x,y)=(x−y)2
Determine whether it is a metric or not.
Solution: d1(x,y)=(x−y)2 satisfies the condition that distance function is positive for all x, y and 0 when x = y since square is always non negative.
Also d1(x,y)=d1(y,x) as (x−y)2=(y−x)2
Finally we check the triangular inequality. We have to show that (x−y)2+(y−z)2≥(x−z)2 . If this is true then we will have 2y2−2xy−2yz+2zx≥0⟹(y−x)(y−z)≥0 . But this is not true for all real numbers. For example if we have x < y < z then (y - x)(y - z) becomes negative. Thus x = 2, y = 4 and z = 7 violates the triangular inequality as (2−4)2+(4−7)2≤(2−7)2.
Hence d1 is not a metric.
(b) d2(x,y)=√|x−y|
Determine whether it is a metric or not.
Solution: d2(x,y)=√|x−y| satisfies the condition that distance function is positive for all x, y and 0 when x = y since square root of |x-y| is always non negative and equals 0 only when x=y.
Also d2(x,y)=d2(y,x) as √|x−y|=√|y−x|
Finally we check the triangular inequality. We have to show that √|x−y|+√|y−z|≥√|x−z|
For x∈R1 and y∈R1 define
(a) d1(x,y)=(x−y)2
Determine whether it is a metric or not.
Solution: d1(x,y)=(x−y)2 satisfies the condition that distance function is positive for all x, y and 0 when x = y since square is always non negative.
Also d1(x,y)=d1(y,x) as (x−y)2=(y−x)2
Finally we check the triangular inequality. We have to show that (x−y)2+(y−z)2≥(x−z)2 . If this is true then we will have 2y2−2xy−2yz+2zx≥0⟹(y−x)(y−z)≥0 . But this is not true for all real numbers. For example if we have x < y < z then (y - x)(y - z) becomes negative. Thus x = 2, y = 4 and z = 7 violates the triangular inequality as (2−4)2+(4−7)2≤(2−7)2.
Hence d1 is not a metric.
(b) d2(x,y)=√|x−y|
Determine whether it is a metric or not.
Solution: d2(x,y)=√|x−y| satisfies the condition that distance function is positive for all x, y and 0 when x = y since square root of |x-y| is always non negative and equals 0 only when x=y.
Also d2(x,y)=d2(y,x) as √|x−y|=√|y−x|
Finally we check the triangular inequality. We have to show that √|x−y|+√|y−z|≥√|x−z|