OPEN SPHERES AND CLOSED SPHERES #open #spheres #sets #metric #spaces #closed
Contents :
1.
Open Sphere
2.
Open sets
3. Results on open sets
Closed sphere
cl Closed set
Results on closed sets
OPEN SPHERE :
Let X be a metric space with the metric d. If x0 is a point of x and r be a positive real number. The open sphere Sr(x0) with center x0 and radius r is the subset of X defined by Sr(x0) = {x ꜪX/d(x,x0) < r}.
NOTE :
1.
An open sphere is always non-empty ,as it contains
its center.
2.
Sr(x0) is often called the
open sphere with radius r centered on x0.
3.
An open sphere consists all points in X which
are “ close “ to x0, with the degree of
closeness given by r.
4.
It should be easy to visualize the open sphere Sr(x0)
on the real line , it is bounded open interval
(x0-r, x0+r) with midpoint x0 and total
length 2r.
5.
Any bounded open interval on the real line is an
open sphere, so the open spheres on the real line are precisely the bounded
open intervals.
6.
The open sphere Sr(x0) in
the complex plane is the inside of the circle with center z0 and
radius r.
The open spheres in the complex plane are often called the open balls.
OPEN SETS :
A subset G of
the metric space X is called an open set If, given any point x in G, there exists
a positive real number r such that Sr(x) ⊆ G.
NOTE :
1.
G is open if each point of G is the center of
some open sphere contained in G.
2.
On the real line, a set consists of a single
point is not open, for each bounded interval centered on the point contains points
not in the set.
3.
Since in any closed interval [a,b], the
endpoints i.e. a and b are not interior points of [a,b], [a,b] is not open.
4.
[0,1) is not open because the left end point 0
is not interior point of [0,1).
5.
Any open interval which is bounded or not is an
open set.
RESULTS ON OPEN SETS:
1. In any metric space X, the empty set ⲫ and the full space X are open sets.
2.
In any metric space X, each open sphere is an
open set.
3.
A subset G of a metric space X is open iff it is
a union of open spheres.
4.
Let X be a metric space. Then
i. Any union of open sets in X is open in X and
ii. Any finite intersection of open sets in X is
open in X.
5.
Every non-empty open set on the real line is the
union of a countable disjoint class of open intervals.
Closed sphere :
Let X be a metric space with the metric d. If x0 is a point of x and r be a positive real number. The closed sphere Sr[x0] with center x0 and radius r is the subset of X defined by Sr[x0] = {x ꜪX/d(x,x0) ≤ r}.
Closed set :
A non-empty set A in a metric space X is said to be closed if A contains all of its limit points.
In the other sense A is called closed if its complement A' is open.
Results on closed sets:
1. In any metric space X, the empty set and the full space X are closed.
2. Let X be a metric space. A subset F of X is closed iff its complement is open.
3. In any metric space X , each closed sphere is a closed set.
4. Let X be a metric space X. Then i) any intersection of closed sets in X is closed
and ii) any finite union of closed sets in X is closed.
people also ask:
1. What is metric ?
3. What is norm of an element?
4. What is norm of a function?
5. What are called interior points?
7. What is an interior of a set?
#open #spheres #sets #metric #spaces
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