Mth634 Assignment solution fall 2021

Mth634

Assignment no. 2

Fall 2021

Disclaimer :- if you find any mistake please correct it by yourself.

 Question: 1

Solution :-

d(x,y)=|x|+|y|

 :d(x,y)=|x|+|y| 0, x,y R

 :We have to show this

d(x,y)=0

|x|+|

|x|=0,|y|=0

x=0,y=0

x=y

then

suppose that x=y

and x 0, y 0

so d is not a matrix on R

Question : 2

Solution :-

X= (a,b) and Y= (0,1) be two subspaces of R with the usual topologies

  is define by

And f(a)=0

f(b)=1

Now we prove that

To show that X and Y are homophormic we will prove that :

1): f is bijective

2): f is continuous

3): f is continuous

As

f is bijective if f is one -to-one and f is onto

1 ): f is one-to-one

Let:

Hence f is one-to-one

2): f is onto

For each y (0,1)

Then there exist x (a,b)

Hence f is onto

Hence f is bijective

2):f is continuous

For each open set v in Y =(0,1)

Hence f is continuous.

3): f is continuous

To prove that f is continuous we prove that f is open mapping

For each open set U on X =(a,b)

  is open Y =(0,1)

As:

X is homorphic on Y