Question

In: Advanced Math

Euclidean metric) The metric d (( x 1 ,x 2 ) , ( y 1 ,y...

Euclidean metric) The metric

d

((

x

1

,x

2

)

,

(

y

1

,y

2

)) =

√

(

x

1

−

x

2

)

2

+ (

y

1

−

y

2

)

2

on

R

2

generates the standard (product) topology on

R

2

.

Expert Solution

The given metric is

First we prove that the metric topology is contained in the product topology on .

Let be a point in and be an open set of metric topology such that . Since is open, there exists an open ball centred at and .

Now for small , is an open set in product topology which contains and is contained in .

Thus the Euclidean metric topology on is contained in the product topology on .

For the converse, let be a point in and be an open set of product topology such that . Since is open ,there is an open set which is contained in for small enough .

Now observe that the ball centred at with radius is contained in and this ball is an open set in the metric topology and further and so the converse is also proved.

And therefore, the metric topology on is same as the product topology on .

Colby Messinger answered 2 years ago

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