1)Prove that the intersection of an arbitrary collection of
closed sets is closed.
2)Prove that the union of a finite collection of closed sets is
closed
Differential Geometry ( Work Shop for Test 1)
(5) Prove that a regular curve (i.e., curve with positive
curvature at all points) is a helix iff the ratio of the torsion to
curvature is a constant. please use Differential Geometry Form not
Calculus.
1. Use the ε-δ definition of continuity to prove that (a) f(x) =
x 2 is continuous at every x0. (b) f(x) = 1/x is continuous at
every x0 not equal to 0.
3. Let f(x) = ( x, x ∈ Q 0, x /∈ Q (a) Prove that f is
discontinuous at every x0 not equal to 0. (b) Is f continuous at x0
= 0 ? Give an answer and then prove it.
4. Let f and g...
Give Examples (this is complex analysis):
(a.) First characterize open and closed sets in terms of their
boundary points. Then give two examples of sets satisfying the
given condition: one set that is bounded (meaning that there is
some real number R > 0 such that |z| is greater than or equal to
R for every z in S), and one that is not bounded. Give your answer
in set builder notation. Finally, choose one of your two examples
and...
"•“"Suppose ℱ, ?1, and ?2 are nonempty families of sets. Prove
that if ℱ ⊆ ?1 ∩ ?2, then ∩?1 ∪ ∩?2 ⊆ ∩ℱ.“"
Please explain what the question is asking for and break down
the solution for me step by step with explanations please!
Provide an example
1) A nested sequence of closed, nonempty sets whose intersection
is empty.
2) A set A that is not compact and an open set B such that A ∪ B is
compact.
3) A set A that is not open, but removing one point from A
produces an open set.
4) A set with infinitely many boundary points.
5) A closed set with exactly one boundary point