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Tuesday, 14 May 2013

Is it a metric?

Q11. Basic Topology (Rudin)

For xR1 and yR1 define

(a) d1(x,y)=(xy)2

Determine whether it is a metric or not.

Solution: d1(x,y)=(xy)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 (xy)2=(yx)2

Finally we check the triangular inequality.  We have to show that (xy)2+(yz)2(xz)2 . If this is true then we will have 2y22xy2yz+2zx0(yx)(yz)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 (24)2+(47)2(27)2.

Hence d1 is not a metric.

(b) d2(x,y)=|xy|

Determine whether it is a metric or not.

Solution:  d2(x,y)=|xy| 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 |xy|=|yx|

Finally we check the triangular inequality. We have to show that |xy|+|yz||xz|
 

Interior of a set

Q9. Basic Topology (Rudin)

Let E0 denote the set of all interior points of a set E. 

(a) Prove that  E0 is always open.

Solution: Suppose x is an arbitrary point in E0. We will show that x is an interior point of E0.

By definition of E0,  x is an interior point of E. Hence there exists a neighborhood N of radius r of x, such that NE.

Since N is a neighborhood therefore N is open. Hence all points of N are interior points of N i.e. they have neighborhoods that are subsets of N. Since N is a subset of E, hence those neighborhoods (of the points of N which are subsets of N) are also subsets of E. Hence all the points in N are interior points of E. Thus all the points of N are member of E0 by definition (since E0 contains all the interior point of E). Thus N is a subset of E0. Since x is an arbitrary element of E0, and it has a neighborhood which is a subset of E0, hence x is interior point of E0 and E0 is open.

(b) Prove that E is open if and only if E0=E

Solution: If E is open, all point of E are interior points of E (by definition). Hence all points of E are in E0 and no other (as E0 contains only the interior points of E and all points of E are interior points). Hence E0=E

Again if E0=E , E contains only the points present in E0 and E0 contains only the interior points of E. Thus all points in E are interior points. Hence E is open.

(c) If GE and G is open, prove that GE0

Solution: Since G is open, all points of G are interior points of G. Hence each point of G has neighborhood N which is a subset of G. As G is a subset of E, hence these N's are also subsets of E. Therefore for each point in G, there exist neighborhood N which is a subset of E. Hence all points in G are interior points of E. Since E0 contains all interior points of E, E0 contains all points of G. Hence GE0.

(d) Prove that the complement of E0 is the closure of the complement of E.

Solution: Complement of E0 has the points which are not interior points of E. Either they are not members of E which makes them member of Ec or even if they are members of E, they are points whose   every neighborhood contains some elements which are not member of E. In other words complement of \whose every neighborhood contains some elements from Ec. Hence they are limit points of Ec.
Thus the complement of E0 contains elements of Ec and limit points of Ec which makes it by definition closure of the complement of E.

(e) Do E and ˉE always have same interiors?

Solution: No. Counterexample: E= Set of Rational Numbers. Interior of this set is Null Set for if q be any rational number, any neighborhood of q will contain irrationals hence it cannot be totally inside E. Limit Points of this set E consists of rationals as well as irrationals. Hence closure of the set of rational numbers is ˉE (set of real numbers). ˉE has plenty of interior points (in fact all points are interior points). Hence E and ˉE does not have same interior.

(f) Do E and E0 always have the same closures?

Solution:

Monday, 13 May 2013

Algebraic numbers are countable

Q2. Basic Topology (Rudin)

A complex number z is said to be algebraic if there are integers a0,a1,,an , not all zero, such that

 a0zn+a1zn1++an1z+an=0 .

 Prove that the set of all algebraic numbers are countable. Hint: For every positive integer N there are only finitely many equations with n+|a0|+|a1|++|an|=N

Solution: 

Set A is countable if we can establish one-one correspondence between A and set of positive integers \(\mathbb{N}\)

By Fundamental theorem of Algebra  a0zn+a1zn1++an1z+an=0  has exactly n complex roots.

Empty Set is a Subset of Every Set

Definition of 'subset' is: A is a subset of B if xAxB
This is an implication statement (If P then Q where P and Q are propositions).
The truth table is Implication Statement is:
P   Q   P=>Q
T   T       T
T   F       F
F   T       T
F   F       T
where T is 'True' and F is 'False'.
That is the truth value of an implication statement is always 'True' when the first proposition (P) is false.
Consider the statement xAxB
Suppose A is an empty set. Then 'x is in A' is false as A contains no element. Hence the implication statement's truth value is True.
Thus A is a subset of B. Since B is any set and A is an empty set, therefore empty set is a subset of any set.