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.
Prove or provide a counterexample
If f is T_C−T_U continuous, then f is T_U−T_C continuous.
Where T_C is the open half-line topology and T_U is the usual
topology.
Prove that the discrete topology on X is the same as the metric
topology induced by the discrete metric.
Where metric topology is defined as:
If (X,d) is a metric space, then consider the collection T of
all open subsets of X. Then (X,T) is topological space. This
topology is called the metric topology on X induced by d.
Let X be a compact space and let Y be a Hausdorff space. Let f ∶
X → Y be continuous. Show that the image of any closed set in X
under f must also be closed in Y .
State a complete proof or disprove with an explicit
counterexample:
a) Every stable matching is also Pareto optimal.
b) Every Pareto optimal matching is also stable.
Topology
(a) Prove that the interval [0,1] with the subspace topology is
connected from basic principles.
(b) Prove that the interval [0,1] with the subspace topology is
compact from basic principles.