Topology
Prove or disprove ( with a counterexample)
(a) The continuous image of a Hausdorff space is Hausdorff.
(b) The continuous image of a connected space is
connected.
Let f : R → R be a function.
(a) Prove that f is continuous on R if and only if, for every
open set U ⊆ R, the preimage f −1 (U) = {x ∈ R : f(x) ∈ U} is
open.
(b) Use part (a) to prove that if f is continuous on R, its zero
set Z(f) = {x ∈ R : f(x) = 0} is closed.
Prove or disprove with a counterexample the next claims:
(a) The complement of a decidable language is decidable.
(b) The Kleene star of a Turing-recognizable language is
Turing-recognizable.
a)
use the sequential definition of continuity to prove that f(x)=|x|
is continuous.
b) theorem 17.3 states that if f is continuous at x0, then |f|
is continuous at x0. is the converse true? if so, prove it. if not
find a counterexample.
hint: use counterexample
Let (X,dX),(Y,dY ) be metric spaces and f: X → Y be a continuous
bijection. Prove that if (X, dX ) is compact, then f is a
homeomorphism. (Hint: it might be convenient to use that a function
is continuous if and only if the inverse image of every open set is
open, if and only if the inverse image of every closed set is
closed).
5. Provide a counterexample to a false statement. (The statement
might be a “for all,” “there
exists,” or P ==> Q.)
6. Compute the product of two sets.
7. Given a relation (as ordered pairs or as a diagram), determine
the domain, range, and target
of the relation.
8. Given a relation (as ordered pairs or as a diagram), determine
if a pair is in the relation.
9. Convert a relation from a list of ordered pairs to a mapping...