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-) Is the following statements TRUE or FALSE? Prove it or give
a counterexample.
i) If f(x) : Rn → R is a convex function, then for
all α ∈ R, the set {x : f(x) ≤ α} is a convex set.
ii) If {x : f(x) ≤ α} is a convex set for all α ∈ R, then f(x)
is a convex function.
b-) Prove that if x* is a vector such that
∇g(x* ) = 0 and ∇2...
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).